Emergence: from physics to biology, sociology, and computer science

arXiv:2508 .085 5 4 November 2025 Emergence : from physics to biology , sociology , and computer science a Ross H. M cKenzie School of Mathe matics and Phys ics University of Quee nsland Brisbane, Austral ia condensed concepts.blogspot .com r.mckenzie@uq.edu .au Abstract Many systems of inte rest to scientists involve a large number of interacting parts and the whole system c an h ave properties that the indivi du al parts do not. The system is qualitatively different to it s par ts. More is different. I take this no velt y as th e defining characte ristic of an emergent propert y. Many o ther characteristics ha ve been a ssociated with emergence are reviewe d, in cluding universality, order, complexity, unpredictability, irreducibility, diversi ty, self-organisation, discontinui ties, and singularities. However, it has not been esta blished whether t hese characteristics are necessary or sufficient for nove lty. A wide range of exam ples are given to show how emergent phenomena are ubiquitous across most sub-fields of physics and many a r eas of biol ogy and social scie nces. Emergence is central to many of the bi ggest scientific and societal c h allenges today. Emergence ca n b e understood i n terms of scale s (energy, time, lengt h, complexity) and the associated stratification of reality. At each stratum (level ) there i s a d istinct ontol ogy (prop erties, phenomena , pro cesses, entitie s, and effective interactions) and epistemology (theories, concepts, models, and methods). This strati fication of reality leads to semi-autono mous scientific disciplines and sub-disciplines. A comm on challenge is understanding the relationship between emergent prop erties observed at the m acroscopic scale (the whole system) and what is known about the microscopic sc ale: the compone nts and their interactions. A key and profou nd insight is to i d entify a relevant eme rgen t mesoscopic scal e ( i.e., a scale intermediate betwee n the macro - and micro- scales) at whic h new entities emerge and interact with one another weakly. In d ifferent words, modula r structures may emerge at the mesoscale. Key theoretical methods are the devel opment and study of effective t h eories and toy models. Effecti v e theories descri be phenome na at a p artic u lar scale and sometimes can be derived from more microscopic descript ions. Toy model s invo lve minimal degrees of freedom, i n teractions, and pa r ameters. Toy mode ls are am enable to analytical and computational a n alysis and may reve al the mini mal requirements for an emergent property to oc cur. The Ising model is an emblematic toy mode l that elucidates not just critical phe no mena but also key characteristics of a https:// doi .org/10.48550 /arXiv. 2508 .08548 eme rgen ce. Many example s are given from condensed matter physics to illustrate the characte ristics of emergence. A wide r ange of areas of physics are discussed, including chaot ic dynamical sys tems, fluid dynamic s, nuc lear physics, and quantum gravity. The ubiquit y of e merge n ce in other fields is ill ustrated by neural networks, protein folding, and social segregation. An e mergent p erspective matters for scientific strategy, as it shapes questions, choice of resea rch methodologies, pri or ities, and all o cation of resources. Finally, the el usiv e goal of the design a nd control of emergent properties is considered. Table of Contents 1. Introduction and Overvie w ............................................................................................. 3 1.1 Goa ls of this article .................................................................................................... 4 1.2 Ten ke y ideas ............................................................................................................. 5 1.3 An exampl e: language and literature ......................................................................... 6 1.4 Scale s and strata ................................ ......................................................................... 7 1.5 The stra tification of nature and scientific disciplines ...................................................... 8 1.6 The sci entific challenge of emergence ............................................................................. 9 2. Characte ristics of emergent phenomena ..................................................................... 10 2.1 Novelty a s the defining characteristic ............................................................................ 11 2.2 Object ive characteristics: discontinuities, order, universality, diversity, modularity .... 12 2.3 Subject ive characteristics: self-organisa tion, unpredictability, i rr educibility, conte xtuality, complexity, closure ....................................................................................... 17 3. M ethods of investigation................................................................ ................................ 22 3.1 Choice of scales to focus on ........................................................................................... 22 3.2 Connect ing stratum: bottom-up or top-dow n? ............................................................... 22 3.3 Identifica tion of parts and interactions .......................................................................... 23 3.4 Phase diagra ms ................................................................................................ ............... 24 3.5 Phenomenol ogy, effective theories, toy models, and microscopic theories .................. 25 3.6 Other issues ................................................................ .................................................... 28 4. Ising models .................................................................................................................... 29 5. Strati fication of physics ................................................................................................. 33 6. Classical phys ics ............................................................................................................. 35 6.1 Theory of elasti city ........................................................................................................ 35 6.2 Therm odynamics and Brownian motion ........................................................................ 36 6.4 Fl uid dynamics and pattern formation ........................................................................... 37 6.6 Caustic s in optics ................................ ........................................................................... 40 6.7 Chaos theory ................................................................................................ .................. 40 6.8 Sychronisati on of oscillators ................................ .......................................................... 44 6.9 Arrow of time ................................................................................................................. 44 7. Q uantu m-classical boundary ........................................................................................ 46 8. Q uantu m condensed matter physics ............................................................................ 48 8.1 State s of matter .............................................................................................................. 49 8.2 Quasiparti cles ................................................................................................................. 51 8.3 Superconduct ivity .......................................................................................................... 54 8.4 Kondo effect................................................................................................................... 57 8.5 Quantum spin c hains ................................ ...................................................................... 58 8.6 Topologi cal insulators .................................................................................................... 59 8.7 Quantised prope rties ...................................................................................................... 59 9. Classical condensed matter ........................................................................................... 62 9.1 Continuous phase t ransitions in two dimensions ........................................................... 63 9.2 Spin ice s ................................ ......................................................................................... 64 9.3 Soft mat ter ...................................................................................................................... 67 9.4 Spin glasses ................................................................ .................................................... 69 10. Nuclear physics........................................................................................................... 71 11. Elementary par ticles and fields ................................................................................ 76 12. Quantum gravity ................................................................ ........................................ 82 13. Complexity theory ...................................................................................................... 86 14. Chemistry.................................................................................................................... 88 15. Protei n folding ................................ ............................................................................ 92 16. Biology ................................................................ ......................................................... 95 17. Economics ................................................................ ................................................. 103 18. Sociology ................................................................................................................... 110 19. Computer science ................................................................ ..................................... 115 19.1 Neural ne tworks ......................................................................................................... 115 19.2 Large Langua ge Models ............................................................................................ 117 20. Philosophical i ssu es ................................................................ .................................. 121 21. Notes on the history of the concept of emerge nce ................................................. 124 22. Conclusions ................................................................................................ ............... 125 1. Introduction and Over view This art icle clarifies what emergence is, its centrality to physics, its relevance to other fiel ds of scienc e, its role in scientific strategy and priorities, and the philosophical questi ons th at it raises. Eme rgen ce is relate d to the observation that the whole can be qualitatively different from the parts. In other words, a system c o mposed of many i n tera cting parts can have new properties that the individual parts do not hav e. This concept describes ma ny fascinating phenomena at the he art of physics, che mistry, biology, psychology, economics, and sociol ogy. A pi ece of solid gold i s shiny, but a sing le gol d atom is not. Wat er is wet, but a sing le water molecule is not. Wet n ess is a property of many wate r molecules that are in the liquid state. Similarly, the mac roscopic properties that distinguish graphite and diamond (e.g., soft vs. hard, black vs. transparent ) are e mergent. Both diamond and graphit e have i d entical co nstituents : ca rbon atom s. Yet they interact in different ways to collectively produce prop ertie s that t h e indivi dual carbon atoms do not have. On the one ha nd, scientific disciplines are very different from one another. They differ in the questions the y seek to answer, the objects studied, the methods used, concept s developed, and the l evels of certainty possible. On the other hand, w hen considered from an emergent perspect ive, there are common features a cross disciplines. Although the focus of this article is on physics I ment ion sc ientific challenge s in other di scipli nes. Emergence is at the heart of some of the biggest questions in each discipl ine. I identify some of the scales and stratum in the di scipline and how these are associated with sub -disciplines. The stratum can be viewed both in t erms of ontology and epistemology. I characterise a system of interest and i d entify the pa rts and the interactions between those parts. I give example s of e mergent prop erties, phenome na, and entities. Emerge n ce is at the heart of many global issues: climate c hange, epidemics, povert y, mental i llness, economic instability, political polari sation, surveillance, and misinformati on on soc ial media. This is because they concern la rge systems w ith ma ny inte ra cti ng p arts. The whole of ten has novel properties that ar e not anticipate d and are resistant to control. In this sense, they are examples of what social scientists call “collective act ion” or “ wicked ” problems. 1.1 Goals of this artic le Clari fy what emergence i s and wha t its characteristics are. Thi s is necess ary beca use of the diversit y and ambi gu ity of views about what emergence is. Introduce physi cists to the role of emergence in othe r fields i ncluding computer science, biology, a nd sociology. Stimulate grea ter cross-fertilisation between physics and other fields. Illustra te how emergence is central to the biggest questions and challe ng es in the sc ience s and in socie ty. He lp physici sts put their work i n a large r context , appreciati ng s imilarities and differences with ot her fields. Consider some of t he practical implications of emergence for how science is done and for deci ding scientific priorities. Describe some of the philosophical issues associated with eme rg ence. The article aims to be pe dagogical and hopefu lly i ns pirational. Referencing is not comprehe nsive, being selective, and is based on accessibility. More background, detail, and differing pe rspectives can be found elsewhere. S emi- popular books on em ergence have been published by Holl and, 1 Johnson, 2 Kauffman, 3 Laughlin, 4 and Morowitz. 5 Academic books have be en written from the perspective of Complexity s cie nce by Jensen, 6 Economics by Krugman 7 and Schelling 8 , Psychology and Sociology by S awyer, 9 Philosophy by Batt erman, 10 Physics by Bishop, 11 Computer Science by Fromm, 12 and a range of fields in a volume edited by Dav ies and Clayton. 13 There is also an article on Emergent Properties in The Stanford Enc yclopedia of Philosophy, 14 and The Routledge Handbook of Emergenc e, 15 and an ant ho logy of classic art icles. 16 1.2 Ten key ideas Consider a system th at is comprised of many interacting parts, invol v es multiple scales, or undergoes ma ny iterations. 1. Many different d efinitions of em ergence have bee n given. I ta k e the def ining characte ristic of an em erge nt property of a system as novelty, i.e., t h e indi v idual parts of the system do not have this property. 2. Many other c haracteristics have been associated with emergence, such as unive rsality, order, complexit y, unpr edictability, i rr educi b ility, diversity, self-organisation, discontinuities, and singula rities. However, it has not been established w hethe r these characteristics are nece ssary or sufficient for novelty. 3. Eme rgen t phenomena are ubiquitous across scientific disciplines from physic s to b iology to sociol ogy to computer science. Cons equently, discipline bound aries become fuzzy with respect to the techniques used and the relevance of results obtained. Emergence is cent ra l to many of t he biggest scientific and societal cha llenges today. 4. Real ity is stratified. At each strat um (l evel) there is a distinct ont ology (properties, phenome na, processes, entities, and effective interactions) and epistemology (theories, conce pts, models, and methods). This stratification of reality leads to semi- autonomous sci entific disci p lines and sub-discipline s. 5. A common c hallenge is understanding the relations hip betwee n emergent properties observed at a macroscopic sca le and what is know n about a microscopic scale: the compone nts and their interactions. A key and profou nd insight is to i d entify a relevant eme rgen t mesoscopic scal e ( i.e., a scale intermediate betwee n the macro - and micro- scales) at whic h new entities emerge and interact with one another weakly. In d ifferent words, a modular structure may emerge at the mesoscale. 6. Key theoretical me thods ar e the devel op ment and s tudy of effecti v e theorie s and toy models . Effective t h eorie s describe phenomena at a particular scale. Sometimes they can be used to bridge the microscopic (or m esoscopic) and macroscopic sca les. Eff ective t h eories are an accura te representation of reality in ways that toy models are not. Toy models involve mini mal degr ees of free dom, interactions, and param eters. Toy m od els are amena b le to anal ytical and computational analysis and may reveal the minimal requ irements for an eme rgen t propert y to occur. T h e Ising mod el is an emble matic toy mod el t h at elucidates not just critical phe nomena but also key cha r acteristics of emerge n ce. 7. Condens ed matter phy sics elucidates many of the key features and c hallenges of eme rgen ce. Unli ke brains and economies, condensed s tate s of matter are simple enough to be ame nable to detailed and definitive analysis but complex enough to exhibit rich and diverse eme rgen t phenomena. 8. The perspective taken a bout emergence by individ ual sci entists and communities m atters for scientifi c strategy . It will shape questions, cho ice of research methodologies, prioritie s, and al location of resources. 9. An emerge nt perspective that does not privilege the parts or the whole can a ddr ess conte ntious iss ues and fashions in t he humanities and s ocia l sciences, particularly around structuralism. 10. Eme rgen ce i s cen tral to important questions in p hilosoph y includi ng the unity of the scienc es, th e relationshi p between theory and reality, and the nature of consciousness. 1.3 An example: language and literature Characteri stics of emergence can be illustrated with an exam p le given by Michael Polanyi: how lite rature emerges from language (Figure 1). A t each level there are disti n ct components, rules of int eraction, phenomena, and concepts. For example, grammar provides the ru les about how words can int eract to produce sentences. Fi gure 1. Eme rg ence a nd the hierarchy of levels associated with language and literature. The boxes represent different components of a system. The rules for interactions betwee n these compone nts are given next to the arrows. I now illustra te s ome characteristics of eme rg ent propertie s with exam p les from la ngu age and lit erature, or iginally noted by Micha el Polanyi. 17 Literary genres include the novel, drama, poetry, comedy, sat ire, romance, and tragedy. Th e concept of a romantic novel h as no mea ning below the top level. I once lea rn t the basics of French vocabulary and grammar and could rea d rud ime n tary sente n ces. However, thi s kno wledge was not sufficient for m e to conce ive of different genres in French literature. A poem involves lines of verse in a parti cular relationship to each other. The concept of a literary genre may be indepe nd ent of what la nguage it is w ritte n in. It does not depend on details of the alphabet, spelling, and gramma r that are used. To il lustrate this effective approach, I r eturn to literature and l anguage. Understanding a parti cular literary genre does not focus on the alphabet, spelling, or grammar. Rather, the genre ma y b e best unde rstood in term s of larger units s uch as paragra phs and literary devices. Again, grea t literary critics have insights about how to do this analysis. b This exa mple of eme rgen ce i s a menable to a more formal a n alysis through Chomsky’s Universal Gramma r . Another exa mple th at is amenable to e xp lai n ing emergence to non-sc ientists is that of geome try and was discussed by Luisi. 18 Consider how dots can be combined to produce li n es, line s to produce planar shapes, and then solid shapes. 1.4 Scales and strata Central to emergenc e is the idea of s cale . Emergent properties only occur when sca les becom e larger. Scales that are simply defined, and might be called extrinsic, include the number of part s in the system, length scale, time sc ale, or energy scale. A more subtle scale, which might be called intrinsic, is a scale associated with an emergent property. Thi s emergent scale is intermediate between that of t h e parts and that of the whole system, i.e., a mesoscale. Exam p les include the persistence length of a po lymer m o lec u le, th e coherence l ength of a superconductor, the Kondo t emperature for magnetic impurities in met als, and the mass of the W a nd Z bosons that mediate the weak nuclea r for ce. Eme rgen t scales lea d naturally to strata or hi erarchi es resulting from a separation of scales. I prefer to not use the term hie r archies as it might sug gest tha t some levels (strat a) are more import ant or fundamental than others . Figure 1 is an example for language a nd literature. There is stratum associated with different scientific disciplines, as discussed below. Strata also occ ur within individual disciplines, leading to sub-disciplines as will be discussed at many poi nts in this paper. At eac h stra tum or level, t h ere is a distinct ontology (what is real) and epi st emology (how we know and describe reali ty). For ontology, there are distinct phenomena, entities, properties, processes, interacti ons, and relations . For epistemology, there are distinct concepts, theories, model s, and sci enti f ic methods. Th e d istinct n ess of levels means there is some autonomy within each le v el. What happe ns with in the level can be describe d and understood without referenc e to other levels. c An emergent prop erty of some systems is that they a re scale-free, with properties that depend on power laws over some ra nge of scales, sometimes covering many orders of magnitude. In scale-fre e systems there is often fractal or self-s imilar structure where the system a ppears to have t he same structure at each scale. Someti mes as the scales inc rease it is said that the co mplexity of the system increases . F or exam ple, humans are more complex than cryst als which a re more c o mplex t h an atoms. However, defini ng complexity is challenging, as discuss ed below, and i s ambiguous. F or exam ple, as one goes from the sca le of quarks and gluons to a hydrogen a tom, the system gets simpl er in some sense. b This exa mple is not a perfect analogy with physical sys tems in which emergent propertie s spontaneously e merge. Th e exam p le of language and literature involves an external agent who create s th e new ent ities w ithin th e constraints al lowed by the rules. c For exa mple, with r egard t o b iology, in 1945 Novikoff argued “the laws describi ng the unique propertie s of each level are qualitatively distinct, and their discovery require s m ethods of research a nd analysis appropriate to the particular level . ” 19 1.5 The stratificati on of natu re and scientific disciplines Fi gure 2a . The stratification of scientific disciplines. The vertical directi on represents an increa se in scal e (number of atoms, length, or time). Fi gure 2b. The stratification of obje cts of interested to physicists. The re is a sub -discipline of physics associat ed with each stratum ( level). The proce ss of consider ing complex systems in terms of their component parts and inte ra cti ons l eads naturally t o a stratifie d v iew of reality. This illuminates the relationship betwee n different scientific disciplines (Figure 2a ). At eac h level of the hierarchy, th e objects of interest are com posed of ob ject s from the level below. F or example, the molecules s tudie d by chemis ts are composed of t h e at o ms studied by physicists. At lower l evels, ther e is a dec r ease in the degree of complexity of the systems (loosely defi n ed as the number of variable s requ ired to give a complete description of the state of the sys tem). In addition, the relevant time and length scales get sm aller at lower strata. d There i s an on tology and an epistemology a ssociated w ith ea ch strata. The ontology includes objec ts, interactions between objects, and phenomena (states, processes, and propertie s). Th e episte mology inc ludes la ws, concepts, categories, or ganising principles, theories, and met hods of investigation. In s ome sense, t h e ontology concerns thi ngs th at ha pp en in the system under study a nd the epistemology concerns things that happen inside the brains of the investi gators. Sometimes the bound aries between strata are not well-defined. A stratum is well-defi n ed if there is cl osure, a concept that is discussed further below. Wit hin disciplines, ther e are also strata, associated with sub-disciplines. F or example, in biology t here are sub-disc ipline s associated with prot eins, genes, c ells, tissues, organs, organisms, and e cology. In phys ics, there are sub-dis cipli nes ass ociated with eleme n tary parti cles, nucl ei, atoms, molecul es, solids, and f luids. Viewing the disciplines as forming stra ta raises interesting questi ons. Wh at is t h e exact relationship be tween the disciplines? D oes the fac t that physics is on the bo ttom mean it is the d This inc rease in scale is not clear cut. For example, some sub-fields of physics, such as fluid dynami cs, ar e conce rn ed with similar time and length scales as biology. It is just that physics is also conc erned with scales much smaller than all the other di sc ipli n es. most fundam ental discipline? If w e fully unde rstand things on one level, can we act u all y expla in everything on the next higher level? For example, can all of biol ogy r eall y b e expla ined solely in terms of chemistry? Up and/or down? Diagram s of strata often include arrows pointing bet ween adjacent stratum. The direction of these a rrows may be associated with an in crease or decrea se of scale, with causality, or w ith expla nation. In Fi gure 2 a, the arrows point up to re present that entities a nd ph enomena on one le v el eme rge fro m the interactions between the entities on the ne xt lowest level. On the other hand, in te rms of the historical progression of science, the arrows tend to point down, as in Figure 2b. For exampl e, in physics understandings of macroscopic phenomena such as therm odynamics and electromagnetism preceded understanding atoms which in turn preceded understandi ng sub-nucl ear physics. Weinberg claims all the arrows of explanation point down. 20 In Andrew Steane’s pic ture of the explanatory relationship between physic s, ch emistry, and biology, he draws arrows pointing in both directions. 21 The up a rrow is denoted “supports [allows and physic ally embodies the expression of]” and the down arrow is denoted “ena rch es [exhibi ts the struct ur es and beha v iours that make sense in their own terms and a r e possible withi n the framework of].” There a r e subtle issues about the role of causality in connecting the different levels, parti cularly downwards. Consider the s imple example of a gas inside a thermally insulated piston. Coll isions between the mol ecules in the gas determine t h e pressure of the ga s and exert a for ce on the p iston wall. On the other hand, sl owly moving the p iston to reduce the volume of the gas w ith increase t h e ave r age kinetic energy of the molecules in t he gas. Eme rgen ce suggests a sci entific strategy for understanding complex systems, whethe r superconduct ors or viruses. Essential el eme n ts of that strategy include focusing on what experi ments tell us, us ing a multi-faceted approach (a range of expe r imental, theoretical, a nd comput ational methods), and devel op ing and understanding simple models that may capture the e ssential features of a phenomenon. This strat egy has implications for setting priorities, w hether for a n individual scientist (choice of topic and technique) or a funding agency (goals and budgets for different areas). It is best to inve st in a portfolio of complementary approaches that look at different scales, from t h e mic roscopic to the macroscopic. 1.6 The scientifi c challenge of emergence Over the l ast century the scie n tific strategy of reductionism , has been very succe ssful in providing bot h a quantitative description and conceptual unde rst anding of a wid e range of natural ph enome n a. For example, biol og ists break organisms down to cells and then to mem bran es, protei ns, and DNA . Physicists consider atom s as composed of nuclei and ele ctrons, nuclei as composed of protons and neutrons, w hich in t urn are composed of quarks. The a dvances made through scientific reductionism are ma ny, such as dis coveries of genetic information e ncoded in DNA and the m olecular basis of genet ics . The “zoo” of elementary parti cles produced by h igh energy c o llisions in p article accelerators can be understood in term s of just a few elementary particles, quarks and leptons. The structure and excitations of mole cules and th e dynamics of chemical reacti ons c an be explained in terms of the laws of quantum theory. But reductioni sm is only part of the story. I know the rules of che ss but that does not make me a grand maste r. Knowing the constituents of a sy stem of scie n tific interest a nd the laws that describe their interactions does not mean we can understand or pre dict the collective propertie s of the sys tem. Inde ed, this is central to the biggest challenges in the natural and social sciences today. Here are just a few examples. Even i f we know the exact geometrical arrangement of the atoms in a solid, it is e x tremely diffic ult to predict whether the material will be magnetic , a metal, an insulator, or a superconduct or. This is w hy condensed ma tter physics is s o full of surprising discoveries. For m ore than t wo de cades we ha ve known the complete DNA sequence of t h e human genome . In prin ciple, thi s con tains a ll the informatio n necessary to unde rstand all diseases. However, understandi ng how different DNA sequence s (i.e., different genes) lead to different biologi cal properties remains a challenge. Know ing the sequence of a mino acids that ma k e up a speci fic protein molecule does not mean we can predict the biological function of that mole cule. Know ing the prop ertie s of neurons in the brain and their interactions does not mea n we can understand consciousness or what thoughts you are going to have. Knowing the psychologic al profiles of the individual members of a crowd does not mean we ca n predict whether i t will become a violent mob. 2. Characteri stics of emergent phenomena There i s no consensus about what emergence is, how to define it, or why it matters. In John Holland’s book, Emergence : from Chaos to Order , he sta tes that, “Despite its ubiquity and import ance, emergence is an enigmatic, recondite topic, more wonder ed at than analysed… It is unlike ly that a topic as complicated as emergence will submit meekly to a concise defini tion, and I have no such to of fer.” 1 Instead, Hol land focuses on system s that can be describe d by simple rules or laws. The rules generate complexity: novel patterns that are someti mes hard to recognise and to anticipate. e Below I seek to clari fy what so me of the important i s sues and questions are a ssoci ated wit h defini ng emergence. I take a path that is intermediate between the precision of philosophers and the loose discuss ion of eme rg ence by many scientists. In 2006, Deg eut et al. 24 made an exte nsive literature survey of emergence definitions and presented five different representati ve definitions. M y goals are clarity and brevity. e Elsewhere, Holland 22 (page 4) stated that what distinguishes complexity from complicated is eme rgence, which is defined in term s of “ the whole is more than the sum of the pa rts.” This can oc cur in a system with “interactions w here the aggregate exhibits propertie s not attained by summat ion ”, i.e ., the interactions are non - linear. This yiel ds l evels of organisation and hierarc h ies, as em phasised by Herbert Simon. 23 2.1 Novelty as the defining characteristic Consider a system that is composed of many interacting parts or invol v es many scales. If the propertie s of the whole system are compared with th e propert ies of the individual parts, a property of t he whole system is an emergent propert y if it is a property that the individual parts of the syste m do not have . Emergent prope r ties are novel . The system is qualitativel y different fro m it s par ts. More is Different, as advocated by Anderson. 25 Exam ples of properties of a physical system that are not eme rg ent are volume , mass, charge , and numbe r of a toms. The se are additive properties a nd sometimes ca lled resultant properties. The propert y of the whole system is simply the sum of the properties of the pa r ts. f Impli cit in this definition is the concept of scale. Some sort of scale (for exa mple, particle number, l ength, or energy) is used to defi ne what the parts are and thus how many parts the r e are. In physics the scales may be microscopic and macroscopic, and perhaps an i n termediate scale , the mesoscopic. For some systems, particularly early in research programs, what the mesoscal e of interest is may not be at all obvious. A subtle questi on is how many parts a system must have (i.e., how large it need be ) for the system to ha ve an emergent property. Cole man 26 foll ow ed A nderson ’s argument 27 , consideri ng small collections of atoms of gold and niobium. A s ingle atom of gold is not shiny but large metallic grains are. A smal l nu mber of niobium atoms are not superconduct ing, only a bulk sample is. K ivel son an d Kivelson 28 claim ed that emergent propertie s can only be defined in the thermodynamic limit. An alte rnative definition is th at an emerge n t property is one tha t occurs in some states of the system but not in a state where the states of the parts are randomly assigned. This means that for physical systems in thermodynamic equilibrium, an eme rg ent propert y is one that the system does not ha ve at high temperatures. The de finition for emergence given above is formulated with physical system s in mind, parti cularly with many particles or degrees of freedom . In this article, I consider a broader range of system s (inc luding social systems, artificial systems, and simple mathematical model s) and so a broad er sense of “many parts” a nd “inte r acting.” Specifically, I include the case of m any iterations, as occurs in dyn amical syste ms and computer algorithms. Kadanoff empha sised how “ th e many times repeated applicati o n of quite sim p le laws ” can l ead to rich new structure s. 29 Examples include cellular automata such as Conway’s Game of Life . A deep ne ura l net work can learn to identify and classify objects, after being trained on a set of many obj ects. For Large Language Models (LLMs) t he “size of the system” can be defined as the run t ime, the size of the data set used for training, or the numbe r of p arameters used in t h e model . 30 An alte rnative way of looking at emergence in terms of novelty is where there is a qualitative difference bet ween local and gl ob al properties. This is helpful when consi d ering dynamical systems, such as those tha t exhibit chaotic dynamics. I don’t claim th at defining emergence in terms of novelty is necessarily better than definitions used by others. I won’t ca talogue or cr itique in detail alternative definitions here. This defini tion in terms of novelty is concrete and precise enough to hel p clarify other f Strict ly speaking, mass is not addi tive if you allow f or E=mc 2 , since the binding energies betwee n particles changes the total mass. characte ristics often associated with emergence. These characteri st ics are sometimes included in the definition of emergence by o ther authors. I have (mistakenly) done this in the past, in blog posts, tal ks, papers 31 , and in my recent Very Sh ort Introducti on to condensed matter physics. 32 There i s more to emergence than novel proper ties, i.e., where a whole syste m has a propert y that the individual components of the system do not have. Here I focus on emergent propertie s, but in most cases “property” might be replaced with state, phenomenon, process, or entit y. I now discuss twelv e characte r istics, besides novelty, t hat are sometimes associated with em ergen ce . S ome people include one or more o f these characte r istic s in their definitions of emerge n ce. However, I do not include them in my definition because some may not be nece ssary or sufficient for novel system properties. g It ma y be helpful to think of the problem of defi n ing the concept of emergence in terms of the philosophic al idea of fam ily resem b lance, made pop ular by L udwig Wittgenstein, and describe d in h is posthumously publi shed book Philosophical Investigations (1953) , §65-71. He argued t hat something tha t could be thought to be defined by one essential fea ture may in fact be connec ted by a series of overlapping similarit ies, where no one fe ature is common to all of the things. He i llustrated his a rgu ment wit h the example of what is a “game.” The fi rst set of six characteri st ics discussed bel ow m ight be classified as objective (i.e., observable proper tie s of the system) and the second s et as subjec tive (i.e., associated with how an investi gator thinks about the system). In different words, t he first set are mostly conce rned with ontology (wha t is real ) and the second set with epi st emol ogy (wha t we know). The first set of c h aracteristics concern discontinuities, structure, modification of parts, universal ity, diversi ty, mesoscales, and structure. The second set concern s elf -organ isation, unpredictabi lity, irr educibility, downward causati on, complexity, and closure. Some exam ples w ill be g iven t o illustrate how it is possible to have novelty wit h or wi thout some of these c haracteristics. 2.2 Objecti ve characteristics: discontinuities, order, universality, diversity, modulari ty 1. Discontinui ties Quantit ative changes in the system can become qualitative changes in the syste m. For exam ple, in condensed matter physics spontaneous symmetry breaking only occurs in the therm odynamic limit (i.e., w hen the num b er of particles of the system b ecomes infinite). More is different . Thus, as a quantitative c h ange i n the sys tem size occurs the order paramete r becomes non-zero. In a sys tem t h at undergoes a phase t ransition at a non-zero tem perature, a small change in temperature can lead to the appearance of order a nd to a ne w state of matter. For a first-order phase transition there is discont inuity in prop erties such as the e ntropy and density. These discontinuities de fine a phase boundary in the pressure- tem perature d iagram . For continuous phase transitions the order paramete r is a continuous function of tem perature, becoming non- zero at the critical temperature. But the derivative with respec t to temperature may b e discontinuous and/or thermodynamic properties such as the spec ific heat and the sus cepti b ility associated with the orde r parameter may approach g In discussing eme rgence in economics, Lewis and Harper 33 pointed out the diversity of perspect ives of what eme rg ence is and sought to clar ify the relationship between novelty, unpredictabi lity, and irreducibility. infini te as the critical temperature is approached. N ew s tate s of matter, including liquid crystal s, antiferromagnetism, and superfluid 3 He, ha ve been discovered by observing disconti nuities in thermodynamic quantities. A discontinui ty is not necess aril y equivalent to a qua lit ative difference (novelty). This can be ill ustrated with the case of the liquid-gas transition. On the one hand, t h ere is a disconti nuity in the density and entropy of the syste m as the liquid-gas phase boundary is crossed in pressure-tempe rature diagram. On th e other hand, there is no qualitative difference bet we en a gas and a l iquid. Ther e is only a qua n titative difference: the density of the gas is less than the liqui d. Albeit sometimes the difference is orders of magni tude. The liquid and gas state can be adi abatically connected, i. e., one can smoothl y d eform one state into the ot h er and all the propertie s of the sys tem also change smoothly. Specifically, there is a pat h in the pr essure- tem perature phase diagram that can be followed to connect the liquid and gas states without any discont inuities in properties. The ferromagne tic state also raises questions about the relationship between novelty and disconti nuities. Th is was illustrated by a debat e betw een Peierls 34,35 and Anderson about whether fe rromagnetism exhibits spontaneous symmetry breaking. On the one hand, singulari ties in properties at the Curie temperature (c ritical tempe r ature for fe rro magnetism) only exi st in the thermodynamic limit. Also, a small change in the temperature, from just above t he Curie temperature to below, can produce a qualitative cha ng e, a non -zero magne tisation. Anderson argued that f erroma gn etism did not involve spontaneous symmetry breaking, a s in contrast to the antiferromagnetic state, a non-zero magnetisation (order paramete r) occurs for finite systems at zero temperature and the magnetic order does not change the excitation spectrum, i.e., produ ce a Goldstone boson. Some condensed matter systems exhibi t novelty without discon tinuities a s the temperature is lowered. Instead, there is a smooth c rossover. Exa mples include spin ice s, Kondo s ystems, Fermi liquids, and glasses. In social sciences, the term tipping point is often used to refer to a qualitative c h ange produced by a qua n titati ve change. 36 2. Order and Structure Eme rgen t properties are often associated wit h the state of the system exhi b iting patterns, order, or structure , terms t hat may be used interchangeably. This reflects that there is a parti cular relationship (correlation) between the parts which is differe n t to the relationships in a stat e without the em ergent prope r ty. In condensed matter physics, the order and structure is often a ssociated with symmetry breaking, which An derson considered t o b e the organisat ional principle for understanding the stratification of reality. 25 Symmet ry breaking is associated with a gene r alised rigidit y. For example, appl y ing a force to one surface of a sol id results in all t h e atoms in the s olid experiencing a force and moving toget her. The rigidit y of the solid reflects a particular relationship between the parts of the system. Anderson state d 27 (p.49) that we “are so accustomed to t h is rigidity propert y that we don’t ac cept its almost miraculous nature, that it is an “eme rg ent propert y ” not c on tained in the sim ple laws of physics, although i t is a consequence of them.” He argued that g eneralised rigidi ty was responsible for most of uniqu e properties of ordered states. 3. Modifi cation of the parts Propertie s of the individual parts may be different in an emergent state. In a state of matter, where the re is an order pa rameter associated with sy mme try breaking , this modi f ies local propertie s. For example , in a ferromagnet, the probability of the magnetic moment of an atom pointi ng in a speci f ic direction is differe n t from i n the paramagnetic state, where a ny direction i s equally likely. In a superconduct or, the local density of electronic states is different fro m in t h e metallic state . Thus, e mergence is associated with novel propertie s a t both the macroscopic and microscopic level. de Haa n refer to the novel prope r ty at the lower leve l as the “conjugate” of the property at the higher leve l. 37 In a crysta l single-a tom properties such as elec tronic energy le v els cha nge quantitatively compa red to their val ues for isolated atoms. Properti es of finite subsys tems a r e also modified, reflecti ng a change in interactions between the parts. For example , in a molecular crystal the frequencies associa ted with intramolecular atomic vibrations are different to their values for isolat ed molecules. However, emergence is a sufficient but not a necessary condition for these m odifications. In gas and liquid states, novelty is not present but there are still chang es in the proper tie s of th e individual pa r ts, such as the vibrational frequencies of m o lecules. 4. Universalit y Many deta ils do not matter. There a r e several dimensions to universality. They are related but not nec essarily equivalent. Many different systems exhibit the same phenomenon. Different phe no mena are caused by essentially the same mechanism and so can be described by essentia lly the same theory. For a specific system s ome properties of the syste m are irrelevant to whether or not the system has the emergent property. The pa ra meter depe ndence of some properties of the s ystem have a universal functional form. An exampl e of i. is tha t superconductivit y is present in metals with a d iverse ra nge of crystal structures a nd chemical compositions. In oth er words, iii ., is that the emergent property is indepe ndent of many of the details of the parts of th e s ystem. Robustness is an exa mple of iii. If small changes are made to the composition of the sys tem (for e xample replacing some of the a toms in the system with atoms of different chemical element) the novel property of the system is stil l present. In e lementary superc ondu ctors, introduci ng non-magnetic impurity atom s has no effect on the superconductivity. Universali ty is s ignificant as it provides a basis for effective theories and toy m odels that can describe a wide range of phenomena at one str atum and requires little detailed information about l ower stra ta. Universality also justi fies the cros s -discipli nary fertilisation of concepts and me thods. For example , the concept of spontaneo us symmetry brea k ing provide s an organising pri nciple for understanding phenomena in both condensed matter physics and ele mentary particle physics, even though these sub -discipli n es concern phenomena at length, time, and energy scales that diffe r by many orders of magnit ud e. Universali ty is both a bless ing and a curse for theory . It is a ble ssing because s olving a problem for one specific material can also solve it for w hole fam ilies of materials. This universal ity is also of deep conceptual significance as understanding a general phenomenon is usually m ore powerful than just a specific example. Universali ty can make i t easie r to develop succe ssful theor ies because it means that many deta ils need not be included i n a theory to suc cessfully describe an em ergent phenomenon. Theoretic al analysis becomes tractable. Effective theories and toy models can work even bett er than might be expected. Universality can make theories more powerful because they can de scribe a wider range of s ystems. F or example, properties of elemental superconductors can be described by BCS theory and by Ginzburg-Landau theory, even though the materials are chemi cally and structurally diverse. The c urses of universa lity for t h eory are that it increases the problems of “under - dete rmination of theory”, “over - fi tting of data” and “sloppy theorie s” . 38 – 40 A th eory can a gr ee with the experiment even when the par ameters used in the the ory may be quite d ifferent fro m the actual on es. For example , the observed phase diagram of water can be reproduced , someti mes w ith im pr essive quantitative d eta il, by co mbining cla ssical statistical mechanics with em pirical force fields that assume water molecules can be treated purel y b eing composed of poi nt charges. Suppose we start with a spec ific microscopic the ory and calculate the macroscopic properties of the system, a nd they a gr ee with e xp eriment. It would then be tempting to think that we have t he correct microscopic theory. However, universality suggests thi s may not be the case. For exampl e, consider the case of a gas of weakly interacting atoms or molec u les. We can treat the ga s par ticles as cl assical or quantum. Statistical mechanics gives exactly the same equat ion of state and specific heat capacity for both microscopic descriptions. The only difference ma y be the Gibbs para dox [ the calculated entropy is not an extensive quantity] which is sensiti ve to whether or not the particles are treated as identic al or not. Unlik e the zerot h, first, and second law of th ermodynamic s, th e third l aw does require that the mic roscopic theory be quantum. Laughlin discussed 4 these i ssues in terms of “ prote ctorates ” that hide “ul timate causes” . For example, the success of elasticity theory which assumes t hat mat ter is continuous hides the underlying d iscrete atomic structure of matter. In some physica l systems universality can be defi ned in a rigorous technical sense, making use of the c oncepts and techniques of the renorm alisation group and scaling. These tec hniques provide a method to perform coarse graining , to deri v e effective theories and effective interactions, and to define universality classes of systems. There a r e al so questions of how universality is related to the robus tness of strata, and the indepe ndence of effective theories from th e coarse graining procedure. 39,41,42 5. Diversity with l imitations Even when a system is composed of a small number of different components and inte ra cti ons, th e large nu mber of possible stable states with qual itatively different properties that the sys tem can have is amazing. Holla nd uses th e term “perpetual novelty” to describe this 1 . Every snowflake is different . Water is found in 18 distinct sol id states. All proteins are composed of l inear chains of 20 different amino acid s. Yet in the human body there are more than 100,000 di ffer ent proteins and a ll perform speci fic bi ochemical func tions. We encounter an inc red ible d iversity of human persona lities, cultures, and languages. A stunning case of diversit y is life on earth. Billions of different plant and animal species are a ll an expression of different linear com binations of the four base pairs of DNA : A, G, T, a nd C. This dive rsity is related to the id ea that "simple models can describe com p lex behaviour". One exam ple is Conw ay’s Game of Li f e . Another example is how simple Ising models with a few compe ting interactions can describe a devil's staircase of ground s tates or the multit ude of different atomic orderi ngs found in b inary alloys. Condensed matter physi cs i llustrates diversit y with the m any differe n t states of matter that have been discovered. The underlying micr oscopi cs is “just” ele ctrons and atomic nuclei interacting according to Coulomb’s law. A typic al game of chess may potentially involve the order of 10 50 move seque n ces. However, game s can be understood in terms of “motifs”: recurring patterns (sequences of move s). There a r e about 50 common motifs. The signi ficance of this dive rsi ty might be downplayed by saying that it is just a result of combi natorics. In a syste m composed of many c omponents each of which can take on a few state s the number of possible states of the whole system grows exponentiall y with the number of component s. But such a claim overlooks the issue of the stabili ty of the diverse states that are observed. For exa mple, for a chain of ten amino aci ds there are 10 13 different possible line ar sequences. But this does not mean that all these sequences will produce a fun cti on al protei n, i.e., a molecule that w ill fo ld rapidly (on the time sc ale of milliseconds) i n to a stable tert iary structure and perform a useful biochemical f uncti on such as catalysis of a speci fic chem ical reaction. h A broader question i s, how does s tability lim it diversity? 6. Modularity at the mesoscale Systems with em ergent properties often exhibit structure at the mesoscale, i. e., interme d iate betwee n the micro- and the macro- scales. Furtherm ore, this struc ture reflects the emergence of entitie s tha t interact weakl y with one another. In o ther words, the sys tem can be vi ewed as a set of i nteracting modules, with each module composed of collecti ons of m icro- constit uents. The a ncient Greeks showed how any mechanical machine could be constructed from six parts: l ever, screw, inclined plane, wedge, wheel, and pulley. A key idea in condensed matter physics is that of qu asiparticle s. A s ystem of strongly inte ra cti ng p articles may have excitations, seen in experiments such as inelastic ne utron scatt ering and Angle Resolved PhotoElectron Spectroscopy (ARP ES), that c an be described as weakly i nteracting quasiparticles. These entitie s ar e composite pa rticles, and have propertie s that are quantitatively d ifferent, and sometimes qualitatively differe n t, from t h e mic roscopic particles. The existence of qu asiparticle s leads nat urally to the t echni que of construct ing an effective Hamiltonian [ effective theory] for the system where effective inte ra cti ons descr ibe the inte r actions between the quasiparticle s. The e conomist Herbert Simon argued that a characteristic of a complex system is that the system c an be understood in terms of nearl y d ecomposable unit s . 23,43 He identified hierarc h ies as an e ssential feature of complex systems , both natura l and artificial. A key property of a level in the hierarchy is that it is nearly decom posable into smaller units, i.e., it can be viewed as a collection of weakly interacting un its. The time required for the evol u tion of the whole system is s ignifi cantly decreased due to the hierarchical character. The construct ion of an artificial complex system, such as a clock, is faster and more reliable if different un its are f irst assembled separately and then the units are brought togethe r into the whole. Simon a rgued that the reduction i n time scales due to modularity is why biol og ical evolut ion can occur on realistic time scales. Laughl in et al. argued the me soscale was key to understanding emergence in soft m atter. 44 Rosas et al ., argu ed that eme rg ence is associated with there being a scale at which the system is “strongly l umpable” . D enis Noble has highlighted how biologi cal systems are modular, h Simila r ly give n the large number of genes (each of which can be switched on or off), why are the r e not a n incredi b ly large number of cell type s ? [Lewontin’s pa radox]. i.e., c omposed of simple interchangeable components. 45 Modularit y is also reflected in the moti fs iden tified in c h ess, music, visual arts, literature, and genetics. Identifyi ng the releva nt scale and the corresponding modula r units may be highly non-tr ivial and represent a signifi cant bre akthrough. Examples include atoms, DNA, and genes. The scale may not be spatial but relate to energy, momentum, numbe r of p articles, time, or connec tivity. As stated at the beginning of this section th e six characteristic s abov e might be associated with ontology (what is real ) and objective properties of the system that an investigator observes and depend less on what an observer t hinks about the system. The next six characte ristics are argu ably more subjective, being concerned wit h epistemology (how we dete rmine what we believe is true). In making this dichotomy I do not want to gloss over t h e fuzzine ss of the distinction or two thousand years of philosophical debates about the relationship be tween reality and theory, or be tween ontology and e p istemology. 2.3 Subjective characteristic s: s elf-organisati on, u n predictabil ity, irreducibility, contextuality, c omplexity, closure 7. Self -organisation Self-organisation is not a prop erty of the syste m but a mechanism that a theorist says cause s an em ergent property to come into being. Self-org anisation is al so referred to as spontaneous order. In the social sci ences s elf -organ isati on is sometimes referre d to as an endogenous cause, i n contrast to an exogenous cause, on e that arises from outside t he system . There is no exte rna l force or agent causing the order , in contrast to order that is imposed externally. For exam ple, s uppose that in a city there is no government policy about the price of a loaf of slice d wholemeal bread or on how many loave s th at bakers should produce. It is obs erved that prices are almost always in the r ange of $4 to $5 per loaf, and that rar ely bread shortages occur. Thi s out come is a result of t h e self-organisation of the free-market, and economists would say the pri ce range and its stability has an end ogenous cause. In contra s t, if the governme nt legislated the price range and the production levels that would be an exogenous cause. Distinguishing exoge n eous and endoge nous c auses is crucial to identifying emerge n t phenome na. It is not always obvious as defining the boundary of a system and its environm ent may b e subjective at some level. In physics, the peri od icity of the a rr angement of atoms in a crystal is a re sul t of self- organisat ion and has an endogenous cause. In contra st, the pe riodicity of ato ms in an optical lat tice is dete r mined by the la ser physi cist who creates the lattice a nd so has an exogenous cause. Fri edrich Hayek empha sised the role of spont aneous order in economics. 46,47 In biology, Stuart Kauffma n equate s e mergence with spontaneous order and self-organisation. 48 Self-organisation shows how local interac tions can p roduce global properties. In different words, s hort-range i nteractions ca n lea d to long-range order. After decades of debate and study, the Ising model showed that this was possible. 49 Other exa mples of s elf-organi sa tion incl ude f locking of birds and teamwork in ant colonies. The re i s no dir ector or leade r bu t the system ac ts “as if” there is. A s Adam Smith observed, econom ic sys tem s act “as if” there is “an i nvisible hand” guiding them. 8 Self-organisation does not necessaril y involve order. For example, it may involve the production of a pp arent noise in a deterministic sys tem, s uch as can occur in a achaotic dynami c al system. Prior to the development of chaos theory, many scientists assumed that apparent r andomne ss they observ ed in a sys tem they were studying was exte rnal to the sy stem rather than being intrinsic to it. 8. Unpredictabi lity The bi ologist, Ernst Mayr def ined emergence as “i n a structure d syste m, new propert ies eme rge at hi gh er levels of inte gr ati on that could not have bee n pr edic ted from a knowledge of the l ower- l evel compone nts.” 50 (p.19). Recently, Philip Ball also defined emergence in terms of unpredictabi lity. i , 51 (p.214) . Someti mes unpredictabilit y is expressed as “surprise” or “un - anti cipation” of the observation of a phenomenon. N ote that the se are subje ctive criteria. Broadly, i n di scussions of emergence, “prediction” is used in three differe nt senses: logical prediction, historical pr ediction, and dynamical pred iction. Logical predi ction (dedu ction) concerns whether a scientist ca n pred ict (calc u late) the eme rgen t (novel ) proper ty of the whole system solel y from a knowle dg e of al l the properties of the pa rts of the sys tem a nd thei r interactions . Logi cal predictability i s one of the most conte sted characteristics of emergence. Sometimes “predict” is replace d with “difficult to predict”, “e x tremely difficul t to predict ”, “impossible to predict ”, “alm os t impossible to predict ”, or “possible in princ iple, but impossible in practice , to predict. ” These are not the same t hing. Phi losophers distingui sh be tween epistem ological emergence and ontological eme rgen ce. T hey are associated with prediction that is " possible in pri nciple, but difficult in practice " and "impossible in principle". 52 After an e mergent property has been discovered experimentally sometimes it can be understood in t erms of the properties of the system parts. In a sense “pre - diction” then becom es “post - d iction.” An exam p le is the BCS theory of superconducti v ity, which provided a posteriori , rathe r than a priori , understanding. In different words, development of the theory was guide d by a knowledge of the phenomena that had already be en observed and characte rised experimentally. Th e phenomen on was a “Black swan” , i.e., a n un anticipated event with significant consequences, whose existence is rationalised after i ts occurrence. 53 Thus, a keyword in t he statement above about logical prediction is “solely” . Historical predicti on . Mos t new states of matter di sc overed by expe rimentalists were not predicted e ven though theorists knew the laws that the microscopic components of the system obeyed. E xamples include superconductivity (eleme ntal metals, cuprates, iron pnictides, organic c h arge t ransfer salts, …), superfluid ity in liquid 4 He, anti f erromagnetism, quasicrysta ls, and the intege r and fr actional qu antum Hall sta tes. There a r e a few excepti ons w here theorists did predict new states of matter. These include are Bose-Einstei n Condensates (BECs) in d ilute atomic gas es and topologica l insulators, th e Anderson insulator i n disordered metals, the Ha ldane phase in e ven- integer quantum anti ferro magnetic spin chains, and the hexatic phase in two dime nsions. I not e that prediction i P. Ball, “T he New Math of How Large Scale Order Emerge s, Quant a, J une, 2024, https:/ /www.quantamagazine.org/the-new-m ath-of-how-large-scale-order-emerges- 20240610/ . of BECs and topol ogical insulators were sign ifica n tly helped that theorists coul d pr edict them starti ng with Hamiltonians of non-in teracting pa r ticles. Furthe r more, all t h e successful predictions liste d above involved working with e ff ective Hamiltonians. None started with a mic roscopic Hamiltonian for a material with a specific chemical composition. Dynamical unpredictabilit y concerns what it means in chaotic dynamical systems, where i t relates to sensit ivity to initial conditions. On the one hand, it might be argued that this is not an exam ple of emergence as most of th e systems considered have only a few degrees of freedom. On the o ther hand, the cha o tic behaviour emerges after many iterations and the s tate of the system can be vi ewed as a time series. Fi nally, the unpredictability of emergent phenomena is connected to t h eir resista n ce t o control as will be discussed later. 9. Irreducibil ity and singularities An emerge nt property cannot be reduced to propertie s of the part s, bec ause if emergence is define d in terms of novelty, t h e parts do not ha v e the property. Eme rgen ce i s a lso associat ed with the problem of the ory reducti on. Formally, this is the process where a m ore general theory redu ces in a particular mathematical limit to a less general theory. For exam p le, quantum mechanics reduces to classical mechanics in the limit where Planc k’s constant goes to zero . Einstein’s theory of special r elativity reduce s to Newtonian mechanics in the lim it where t h e speeds of massive objects become m uch less than t he speed of light. Theory reduction is a subtle philosophic al proble m that is arguably poorly understood bot h by scientists [who oversimpl ify or triviali se it] and philosophers [who arguably ove rst ate the problems it presents for s cience produci ng reliable knowledge]. Subtlet ies arise because the two different theories usually i nvo lve language and conc epts that are "incommensurat e" with one another. Irreducibili ty is also related to the di scon tinuities and singulari ties associated with emergent phenome na. As emphasised indepe nd ently by Hans P rimas 54 and Michael Berry 55 , singulari ties occur because the mathematics of theory reduction may involve singular asymptot ic expansions. Primas illustrated this by considering a light wave incident on an objec t and producing a shadow. The shadow is an emergent property, well describe d by geome tric al optics, but not by t he more fundamental theory of Maxwell’s electromagnetism. The t wo theories are related in the asymptotic limit that the wav elength of light in Maxwel l’s theory t ends to zero. Th is example illustrates that theory reduction is compatible with the eme rgen ce of nove lty. P rimas also consider ed how t he Born-Oppenheimer approxi mation, which is ce ntral to solid state theory and quantum chemistry, is ass ociated with a singular asymptot ic expansion ( in the ratio of the ma ss of an electron to t h e mass of an a tomic nuclei in the system). Berry conside r ed 55 several other examples of theory reduction, including going from general to speci al relativity, from statistical mechanics to thermodynamics, and from viscous (Navi er - Stokes) fluid dynamics to i nv iscid (Euler) fluid dynamics. He has discussed in detail how the causti cs that occur in ray optics are an emergent phenomenon and are associated with singular asympt o tic expansions in the wave the ory. 56 The phi losopher of science Jeremy Butterfield showed rigorously tha t theory reducti on occurred for four spec ific systems that exhibited emergence, defined by him as a nove l and robust property. 42,57 Thus, novelty is not sufficient for irreducibility. 10. Contex tuality and downward causation Any real system ha s a context. For e x ample, a physical system has a boundary and an environm ent, both in time and spac e. In many cases the properties of the system are compl etely determined by the part s of th e system and their interactions. Previous history and boundarie s do not matter. H owever, in some cases the context may have a significant influe nce on the state of the system. Ex amples include Rayleigh-Bernard c onv ection cells and turbule nt flow w hose existence and nat ur e are determined by the interaction of t he fluid with the containe r bound aries. A biol ogical example con c erns what fac tors determine the structure, prope r tie s, and function that a p articular protein (linea r chain of amino aci ds) has. It is now known that the onl y f act or is not just the DNA sequence t hat encodes for the am ino aci d sequence, in contradiction to s ome ve rsions of t he Central Dogma of molec ular biology. 51 Other factors may be the t yp e of cell that contai ns the protein and the n etwork of other proteins in which t h e particul ar protein is embedded. Context sometimes matters. Eme rgen t properties are not determined solely by th e mic roscopic components of the system and thi s has led to the idea of contextual emergence. 58 Supervenie nce is the idea t h at once the micro level is fixed, macro le v els are fi xed too. Th e exam ples above might be interpreted as evidence against supervenie n ce. S upervenience is used to argue against “the possibility for mental causation above and beyond physical causat ion.” 59 Downward causation i s sometimes equated with emergence, par tic u larly i n d ebates about the nature of c onsc iousness. In the conte x t of biology, Nob le defin es downward causati on as when higher l evel processes can cause changes in lower level prop ertie s and processes 45 . For exam ple, phys iological effect s c an switch on and off individual genes or signalling processes in ce lls , as occurs with maternal effects and e p igenetics. 11. Complexi ty Simple rules can lead to complex behaviour. This is nicely illustrated by cellular a u tomata. 60 It is al so seen in other systems with emergent properties. For e x ample, the laws describi ng the propertie s of electrons and ions in a crystal or a large molecule are quite simple : Schrodinger’s equa tion plus Coulomb’s law. Yet fro m these sim ple rules, complex phenome na emerge: all of chemistry and condensed matter physics! 61 There i s no agreed universal measure for the comple xity of a syste m with many components. When some one says a p artic u lar system is "complex ” they may mean there are many degrees of freedom a nd /or that it is hard to understa nd. "Com plexi ty" is sometimes us ed as a buzzword, just l ike "emergence." Complexity means different things to different people. In More is Diffe rent , Anderson sta ted 25 that as one goes up the hierarchy of scientific discipl ines the system sc ale and complexi ty increase s. This makes sense when com p lexity is define d in terms of the number of degrees of freedom in the system (e.g., the size of the Hilbert space needed to describe the complete qu antum state of the system or classically, the position a nd speed of all the constituent particles). O n the other hand, from a coarse-grained perspect ive the syste m stat e and its dynamics m ay become simpler as one goes up the hierarc hy. The equilibri u m thermodynamic state of the liqui d can be described c o mpletel y in term s of the density, temperature, and the equation of state. At that level of description, the system is argua bly a lot simpler than th e quantum chromodynamics (QCD) descripti on of a single prot on. Thus, we need to be clearer about what we mean by complexity. In 1999 the j ournal Science had a special issue that focuss ed on com plex systems, with an introduction e ntitled, Be yond Reductioni sm . 62 Eight survey article s covered complexity in physics, chem istry, biology, earth science, and economics. Ladyman et al. 63 po in ted ou t tha t eac h of the authors of these articles chose different properties to define what complexity is associat ed wi th. The different characteristic s chosen included non-linearity, feedback, spontaneous order, robustne ss and lack of central control, emergence, hie rar chical organisat ion, and numerosity. The problem i s that these cha racteristics are not equivalent. If a specifi c definition for a complex system is chosen, th e diffic u lt problem t h en remains of dete rmining whether each of the characteristics abov e is necessary, sufficient, bot h, or n eit h er for the system to be complex as defined. This is simi lar with attempts to de f ine emerge nce. Ladym an et al. have systematically stud ied d ifferent definitions of a complex system. They compa re d ifferent qu antitative measures such as those based on informati on content (Shannon entropy, Kolm ogorov complexity), de terministic complexity, and statistical compl exity. 63,64 They distinguish different measures of complexity. There are three distinct target s of measures: methods used, data obtained, an d the system itself. There are three types of mea sures: difficulty of description, difficulty of cr eat ion, or degree of organisation. To com plicate matters more, the effective theories that describe many emerge n t phenom ena are simple in t h at the rele v ant equations are simple to write down and involve just a few varia bles and param ete rs. Th is contrasts with the underlying theories that may involve many degrees of free dom. For example, compare elasticit y theory with t h e dynamical equa tions for all the atoms in a crystal. In other words, simplicity emerges from complexity. 12. Intra-stratum closure Rosas et al . re cently conside red emergence fro m a computer science perspect ive 65 . They defined emerge nce in t erms of universalit y and discuss ed it s relationship t o infor mational closure, ca usa l cl osure, and computational closure . Each of these are given a precise technical defini tion in their paper. H ere I only give the sense of their de f initions. In considering a general system they do no t pre-define the micro- and macro- level s of a sys tem but consider how they mi ght be defined s o that universality holds, i.e., properties at the macro-level are indepe ndent of the d etails of the micro-level. Informational c losure m eans that to predict the dynamics of the system at the macrosca le an observer does not nee d any additional information ab out the details of the system at the mic roscale. Equ ilibrium thermodyna mics and fluid dynamics are examples. Causal closure mea ns th at the system can be controlled at the macrosca le without any knowledge of l ower-l evel i nfor mation. For example, changing the software code that is running on a c omputer re liable control of the microstate of t h e hardware of the computer regardle ss of what is happening with the trajectories of individual electrons in the computer. Computational closur e is defined in terms of “a conceptual device called the ε -m achine [originally i ntroduced by Shalizi and Crutchfield 66 ]. This device can exist in some finite set of state s and can predict its own future state on the basis of its current one... for an emergent system tha t is computationally closed, the machines at each level ca n b e constructed by coarse-grai n ing the c o mponents on j ust the level below: They are , `strong ly lumpable. ’” The lat ter property is similar to the characte r istic of modularity at the mesoscale. Rosas et al . show ed that informational closure and causal closure a r e equivalent and that they are more r estric tive than computational closure. It is not clear to me how these closures relate to novel ty as a definition of emergence. In the rest of this article, I will discuss emerg ence in a range of scientific disciplines. Foc ussing on some specific scientific problems I ide ntify some of t h e charact eristics discussed above such a s novel ty, discontinuities, universalit y, and modularity at the mesoscal e. The discussions are illustrative no t exhaustive. First, I consider how these characte ristics s hape diffe r ent methods of scientific investigation of systems with emerge n t propertie s. 3. Methods of investigation The e lements are choice of scale to focus on, d ifferentiation and i n tegration of parts, episte mology, effective theories, toy models, intellectual synthesis, discovering new systems, devel oping n ew experiment al probes, and interdiscip linarity. 3.1 Choice of scales to focus on A key is to dec ide at what scales a system of int erest should be studied at, and then to use appropriate m ethods for each of t h ese scales. There a re often t hr ee sca les of interes t: micro, meso, and m acro. What each of these scales a re for a s pecifi c system may no t be obvious, parti cularly in the early stages of an investigation. Even when they are clearly defined and agreed upon c h aracterising a system at one of these s cal es requires multiple approaches and init iatives. Th e choice of scales shapes the choice an d devel opment of too ls and methods, both expe rimental and theoretical, that can be used to study the system. There can be ambiguity when it comes to choosing experimental probes to explore properties at a particular scale. On the one hand, nucl ear physics is irrele vant to cond ensed matter physics, chem istry, and mol ecular biology. Nuclei can be viewed as merely point cha rg es at the c entre of a toms. Nevertheless, experimental meth ods based on physics at t he nuclear scale , such as nuc lear magnetic resonance (NMR) , isotope labelling, and Mossbauer spectroscopy, ha ve b een fruit fu l in these fields. In some cases, these methods are relevant beca use of subtl e effects such as when phenomena at the at o mic sc ale slightly modify electric fiel ds at the nuclear level. 3.2 Connecting stratum: b ottom-up or top -down? A major goa l is to understand the relati onship betwe en differe n t strata . Before describing two alt ernative approaches, top-dow n and bottom-up, I need to point out that in different fields th ese terms are used in the opposite sense. In this arti cle , I use the same terminology tha t is traditiona lly used in condensed matter physics, chemistry, 67 in biol ogy. 68 It is also consistent with the use of the term “downward causation” in philosophy. Top-down mea ns going f ro m long dista nce scales to short distance scales , i.e., going down in the dia gr ams shown in Fi gur e 2. In contrast i n qu antum f ield theory of elementary particles and fields, or high-energy physics, “top - down” means the opposite, i.e ., going from short to long distance length scale s. 69 This is because practitioners in that field tend to draw diagrams with high energies at the t op and low energies at the bottom. Bott om-up approa ches aim to answer the qu estion: how do properties observed at the mac roscale emerge from the microscopic properties of the system ? H istory suggests that this question ma y of ten be best addressed by ide n tifying the relevant mesoscal e at which modula rity is observed and connecting the micro- to the meso- and connecting the meso- to the m acro. Top-down approach es try to surmise something about the microscopic from the macroscopic. This has a long a nd fru itful h istory, albeit proba b ly with many false starts that we may not hear about , un less we live through them or read history books. K epler' s snowflakes are an earl y ex ample. 70,71 Before people were completely convinced about the existe nce of atoms, the study of c rystal facets and of Brownian motion provided hints of the atomic structure of mat ter. Planck deduced the existence of the quantum from the thermodynamics of black -body radiation, i.e ., macrosc opic properties . Arguably, the first definitive determination of Avogadro's numbe r was from Perrin's experiments on Brownian motion which involved mac roscopic measurements. Comp aring classical statisti cal mechanics to bulk therm odynamic prop ertie s gave hints of an underlying quantum structure to reality. The Sackur-Tet rod e equation for the entropy of an ideal gas hint ed a t the quantisation of phase space. 72 The Gibbs para dox h int ed that fundamental particles are indisti ngu ishable. T h e third law of the rmodynamics hints at quantum degeneracy. P auling ’s proposal for t he structure of ice was based on macroscopic measurements of i ts residual e n tropy. Pasteur deduced the chiralit y of mo lecules from observations of the facets in crysta ls of tartaric acid. Som etimes a “top - down” approa ch means one that focuses on the meso-scale and ignores microscopic deta ils. The t op-down and bot tom-up approaches should not be seen as e xclusive or competitive, but rather c o mplement ary. Their relative priority or feasibilit y d epends on the system of int erest and the amount of information and techniques availa ble t o an investigator. Coleman has discussed the i nterplay of emergence and reductionism in condensed matter. 73 In biology, Mayr, 50 (p.20), advocat ed a “ dual level of analysis ” f or organism s. In social science Schell ing, 8 (p.13-14) d iscussed the i n terpla y of the behaviour of individuals and the propertie s of social aggregates. In a classic study of comple x org anisations in business 74 understandi ng this interplay was termed differ entiation and inte gration. 3.3 Identification of parts and interactions To understa nd emergent properties of system a ke y s tep is identifying what the r ele vant parts are in t h e system and this is related t o the c ho ice of the microscopic scale. This is the age nd a of met hodological reductionism and has often been the first step i n massive advances in scienc e. In b iology think of the discovery of c ells, m embrane s, prot eins, and DNA. But understandi ng emergent properties also requires a kn owledge of t he interactions between the parts. There i s a question as to whether pla ce primac y on the parts or their interactions. In his Nobel Prize Lecture , John Hopfield no ted 75 that his 1974 paper on kinetic proof reading in biosynthesis “ was im portant in my approach to biolo gical problems, for it le d me to think about t he function of the structure of reaction networ ks in biology, rat her than the function of the struc ture of the molecules themselves. Six years later I was generalizing this vi ew in thinki ng about networks of neurons rather than the p ropertie s of a single neuron. ” Jensen stated 6 that a distinguishing feature of complexity science is i ts empha sis on th e network of inte ra cti ons. Wh at the parts a re don’ t really matter. I n contrast, consider the quark model for mesons and baryons, originally de veloped by Gell Mann and Zweig in the 1960 ’ s. Primacy was place d on the parts (quarks and their quan tum numbers) not the interactions that held the quarks togethe r. 3.4 Phase diagrams Phase diagra ms ar e ubiquitous in materials scie nce. They show what state s of m atter are therm odynamically stable depending on the value of external parameters such as temperature, pressure, magne tic field, or chemical composition. How ever, t hey are only beginning to be appreciate d in othe r fields. Recently, Bouchaud argued 76 that ph ase diagrams should be used more to unde rstand agent- based models in the soc ial science s. For t heoretical models, whether in condensed matter, dynamical systems, or economics, phase dia grams can show how the sta te of the system predicted by the model ha s qualitatively different prop erti es depending on the parameters in t he mode l, such as the st rength of inte ra cti ons. Phase diagra ms illustrate discontinuities, how quanti tati v e cha ng es produce qualitative change s (tipping points), and diversity (simple models can describe rich behaviour). Phase diagram s show how robust and universal a state is, i. e., whethe r it only exis ts for fine-t uning of parameters. T heoretical phase diagrams can expan d our scient ific imagination, suggesting new regim es that might be explored by experiments. An exampl e is how the pha se diagram for QCD matte r has suggested new experiments, such as at t he Relativistic Heavy Ion Colli der (RHIC). For dynamical systems, this will be illustrated with the phase diagram for the L oren z model. It shows for what param eter range s strange att r actors exist. Today, for the or etical models of strongly correlated electron systems it is common to map out phase dia grams as a functi on of the model parameters. How ever, thi s was not always the case. It was more common to just investigate a model for specific parameter values that were deem ed to be relevant to specific materials. Perhaps, Anderson s timul ated this new approach when, in 1961, he drew t he phase diagram for the mean -field solution to his model for local mome nts in metals, 77 a paper that was partly t h e basis of his 1977 Nobel Prize. At a mi nimum, a phase diagram should show the stat e with the emergent property and the disordered sta te. Diagrams that contain multiple phases may provide h ints for developing a theory for a specifi c phase. For example, for the high -T c cuprate superconductors, the proximi ty of the Mott insulating, pseudogap, and non -Fermi liquid metal ph ases has aided and constra ined theory development. Phase diagrams have al so a ided theory development for superconduct ing organic charge transfer salts. 78 Phase diagra ms constrain theories as they provide a minim u m criterion of somethi ng a successful the ory should explain, even if only qualitatively. Phase diagrams illust r ate the potent ial and pitfalls of mean -fiel d theories. On the pos itive side, they may get qualitative deta ils correct, even for complex phase diagrams, and can show what emerge n t states are possible. G inzburg-Land au and BCS theories are me an-field theories and work extremely well for m any superconductors. On the other hand, in sys tem s with large fluctuations, mean- fiel d theory may fail spectacularly, and these syste ms are somet imes the most interesting and theoretic ally challenging systems. 3.5 Phenomenology, effec t ive theories, toy models, and microscopic theories Wit h respe ct to the theory of eme rg ent phe no mena, people have different views about what theory i s and the criteria used to evaluate the validity , value, or importance of diffe r ent approache s to theory. Often the terms theory and mo del are used interchangeably. In consideri ng the challenges of inter-disc ipli n ary research, Bouchaud distinguished three distinc t models: phenomenological, funda mental, and meta phor ical. 79 . Phenomenological model s s eek t o p arametrise observational data, mostly from the macro-scale. Meta phor ical model s are like the toy models I discuss below. Fund ame ntal models inc lude the effective theori es I discuss and microscopic theory such a s qu antum theory. There i s a common, but not unique, historical sequence in t h e development of the t h eory of many spec ific emergent phenomena. The sequ ence is phenomenol ogy leads t o toy m odels, then e ffe cti v e theories, and finally the derivation (or at lea st j ustification) of the effective theory from microscopic the ory. Although there are exceptions, this history might be helpful in faci ng the challe nge of developing theory for any new s ystem. It should a lso caution about grand schem es to rush towards reductionism (now partly driven by the seductive allure of increa sing computational power). The classific ation be low is not hard and fa st. The boundaries between the types of theory and model ling can b e fuzzy. Phenomenol ogy At the l evel of qua litati v e descriptions phenomenolog y may consist of c lassificati on of systems and thei r prop erties. At the quantitative level, phenomenology may comprise parametri sat ion of experimental data. These em p irical relations provide a concrete challenge for theorists to e xp lain. Phenomenology may suggest what data to gather, how to represent it, and int erpret it . Phenome no logy may provide a way to contrast a nd compare syste ms. Exam ples of empirical relati ons include the power laws observed in diverse sy s tems , 80 the depende nce of the critical temperature of el emental superconductors on the square root of the atom ic mass, the linear in temperature resistivit y of the metallic state of optimally doped cuprate superconductors, and Guggenheim ’ s sca ling plot for liquid-gas critical points. Phenomenol ogical models include Pipp ard’s model for the non -local London equati ons in superconduct ors, the two-level syste ms theory of glass es, and the parton m od el for deep inel astic electron-proton scattering. At best phenome no logy may be a key clue t h at leads to a more microscopic understa nd ing of the phe nomena. At wors t, empirical relati ons m ay be an artefact of c urv e fitting or data selec tion. For example, how many decade s must a p ower law have for it to be sign ifica nt? Eff ective theories An effecti v e theory i s valid at a particular range of scale s. In physics, examples include cla ssical mechanics, general relativity, M axwell ’s th eory of electroma gn etism, and therm odynamics. Th e equations of some effective theor ies can be writte n down a lmost solely from conside ration of symm etry and c onserv ation laws . Example s include the Navier-Stokes equat ions for fluid dynamics and non-l inear sigma models for magnetism. Some effective theori es can be derived by the coarse graining of theories that are valid at a finer scale. For exam ple, the equations of classical mechanics result from taking the limit of Planck’s constant going to zero in the equations of quan tum mechanics. The Ginzburg-Landau t h eory for superconducti v ity can be de rived from th e BCS theory. The parameters in effective theori es m ay be determ ined from more microscopic t heories or from fitting experi menta l data to the pred ictions of the effecti v e theory. Effective theories are useful and powerful because of the minimal ass umptions and paramete rs used in their construction. For the theory to be useful it is not necessary to be able to deri ve the effective theory from a smaller scale th eory, or even to have such a smaller scale theory. F or example, thermodynamics is inc redibly us eful in e ngineering, independently of whether one has an atomic scale description of the m aterials involved. Even though there is no acc epted quantum theory of gravity, general relativity can b e used to de scr ibe phenomena in astrophysic s and cosmology . Effective theories can also describe phenomena at the mesoscal e and form a bridge between the micro- and macro- scales. Toy models In his 2016 Nobel Le cture on Topo logical Quantum Matte r, D uncan Haldane sai d, “Looking back, … I am struck by how important the use of stri pped down “toy m odels” has been in discoveri ng new physics.” 81 Exam ples of toy models incl ud e the Ising, Hubbard, K ondo, Agent-Based Models, Schel ling, Hopfield, and Sherri ng ton-Kirkpatrick models. Most of these models will be discussed in this paper. I refe r to them as “toy” models because t h ey a im to be as simple as possible, while still capt uring the essential details of a phenomena. At th e scale of inte res t they are an approxim ation, neglecting certain degrees of freedo m and interactions. In contrast, at the releva n t scale, some effecti v e theories might be considered exact because they are based on general pr inci p les such as conservation laws. Toy mode ls do not necessarily correspond to any real system. Consequently, specific toy model s, part icularly when the y are originally proposed, have been criticized a s s implistic, misle ading, or of li ttle val u e. Such criticism is more likel y to be found among experi mentalists w ho have spent a lifetime pai nst akingly studying all the d etails of a phenome non and the component s of the underlying s ystem. For example, James Gleick recounts t he resistance that Bernardo Huberm an received when he presented a toy model for patt erns of erratic eye movements ass ocia ted with sc hizophrenia to a group of psych iat rists, biologi sts, and physicians. 82 John Hopfield had a similar experience wit h neuroscientists when he first proposed hi s neural network model for associat ive memory. Historica l experience suggests a strong just ificati on f or the proposal and study of toy models. They a re concerne d wi th qualitative understanding a nd not a quantit ative d escription of experi mental data. A toy model is usually i n troduced to answer basic questions about what is possible . In the c on text of b iology, Servedio et al. 83 r efer to suc h models as “proof of conce pt” models. What a r e the essential ingre d ients that are suffi cie n t for an em ergent phenome non to o ccur ? What details do matter? For example , the Ising model was introduced to see i f it was pos sible for statistical mecha nics to d escribe the sharp phase transition associat ed with ferromagnetism. Holland a rgued that models are essential to understand emergence. 1 Scott Page’s book The Model Think er (and the associated online c ourse, Mo del Thinking ) 84 enumerated the value of simple mod els in the social scie n ces. An earlier argument for their value in biology was put by J.B.S. H alda ne in his semi n al article about “bean bag” genetic s . 85 Simplicity makes toy model s more tractable for mathematical analysis and/or computer simulation. The assumptions made in defi n ing the model can be clearly state d. If the model is tracta b le then the pure l og ic associate d with mathematical analysis lea ds to reliable conclusions. This contrast s with th e qualitative arguments often used in the biological and social sciences to propose expla nations. S uch arguments can miss the counter-intuitive conclusions that are associat ed with emergent phenomena and the rigorous analysis of toy m odels. Toy model s c an show what is possible, what are s imple ingredients for a system s uffici ent to exhibi t an emergent prope r ty, and how a quan titati v e change can lead to a qua litative cha ng e. In different words, what detail s do m atter? In a toy model the ingredients to define are the degrees of free dom, the interaction parameters, and a “cost” function, such as a Hamiltonian, that determines stability, and possibly dynamics. Toy mode ls can provide gu idance on wha t experimental data to gather a nd how to analyse it. Insight can be g ained by consideri ng mu ltiple models as that a pproach can be used to rule out alt ernative hypotheses. 84,86 Fi nally, th ere i s va lue in the ada g e, “a ll models are wrong, but some are use fu l, ” coined by Ge orge Box. 87 All models are wrong in the sense that they do not include all the details of the system, ei ther of the components or their interactions. Nevertheless, they may still describe some phenom ena and provide insights into what is sufficient for a phenomenon to occur. Historica lly, good toy models have been more useful than originally anticipated. Due to unive rsality, sometimes toy models work better than expecte d, and can even give a quanti tative description of experimental data. An exa mple i s th e three-dimensional Ising model , which was found to b e consistent with data on the li qu id-gas transition near the crit ical point. Although, a fluid is not a magnetic system and t h e atoms are not located on a periodi c lattice, the analogy was bolstered by the mapping of the Is ing model onto the lattice gas model . Th is success led t o a shift in the attitude of physicists towards the Ising model. From 1920-1950, i t was viewed a s irr eleva nt to mag netism because it did not d escribe magne tic interactions quantum mechanically. This w as replaced with the view that it was a model that could give insights into collective phenomena, in cluding phase transitions. 88 From 1950-1965, the vie w di minished that the Ising model w as irrele vant to describing critical phenome na because it oversimplified the microscopic interactions . 89 An attra ctive feature of many toy models is that they are amenable to computer simulation beca use the degrees of freedom are discrete variables with only a small num b er of possible value s. This a llows the possibility of c onsid ering zillions of possible confi gurations of systems that cannot be humanly conceived or studied analytically. Th is has driven t he devel opment of Agent-B ased Models in the social sciences. 90 Microscopic theories These sta rt with the constituents at the microscopi c l evel and their inte ra cti ons. For example, in conde nsed matter phys ics and chemistry, constituents are e lectrons and nuclei, and they inte ra ct via the Coulomb interaction. T h e theory is the multi-particle Schrodinger equation with the Hamiltonian: Laughl in and Pines misch ievously dubbed this “The Theory of Everything” as in princi ple it describe s almost all phenomena in chemistry and condensed matter physics. 61 It mi ght b e claim ed that due to the stratified nature of reality any microscopi c theory should be conside red an effective theory. However, what is macroscopic and microscopi c is a subject ive choice. If there is a m esoscopic scale of relevance then the associ ated eff ective theori es m ay help bridge the meso sc opi c and macroscopic, giving an understanding of phenome nology and macroscopic e xp eriments. Microscopic theory provides a justification and possibly a pa rametrisation of effective theories. For example, in condensed matter physics strongly correlate d electron models on a lattice c an be parametrised by electronic structure cal culations based on Density Functional Theory (DFT). 91 3.6 Other issues Intel lectual s ynthesi s When a system has been studied by a range of methods (experiment, theory, compute r simula tion) and a t a range of scales, a challenge is th e s ynthesis of the re sul ts of these different stud ies. Value-laden j udg ements are made about the pri ority, importance, and vali dity of such attempts at synthesis. Often synthesi s is relega ted to a few s ente nces in the introductions and c onclusions of papers. It needs to be more e x tensive and rigorous. New ex perimental probes and s ystems For known systems and e mergent properties, there is the possibility of creating new methods and probes to i nvestigate them at appropriate leng th and time scales. A n exam ple is the devel opment of sma ll angle neut ron scattering for so ft matter. The identification of new mesoscopic (emergent) scales m ay be intertwi n ed wi th the de v elopment of new probes a t the releva n t scale. Sometimes developme n t of the new probe lea ds to discovery of the scale. Other ti mes, identific ation of the scale by theorists motivates the development of the n ew experi mental probe. New systems can be c reated and investigated in the hope of discoveri ng n ew eme rgent propertie s, e.g., n ew state s of matter. A mor e modest, but still important, goal is findi ng a new system that mani f ests a known emergent property but is more amenable to scientific study or tec hnological application. Sometimes advan ces are made by identifying new classes of systems that exhibit an emergent phenomena or n ew membe rs of a known class. Culti vating in ter-disciplinary insights As emergent properties invo lve multiple scales the y are ofte n of intere st to and amena b le to study by more t han one scientific discipline. Differ ent disciplines may contribute compl ementary s kills, methods, motivations, conceptual fra meworks , and practical appli cations. For example, soft matter is of interest to physicists, chemists, biologi sts, and engine ers. Prote in foldi ng whi ch brings together physics, chemi stry, molecular biology, and comput er science. Neural networks brings together physics, neuroscience, psychology, a nd comput er science. Some of thi s interdisciplinarity is underp inned by association of emergence with universality. Two systems composed of very di ff erent c o mponents may exhibit collective phenomena with simil ar characteristics. This creates opportunities and challenges for inter -disciplinary coll aboration. Cross-fert ilisation is often not a nticipa ted. For example, studies of s pin gla sses in conde nsed matter led to new methods and insights for protein folding, neuroscience, comput er science , and evol u tionary biology. The e xtent th at any of the a bove methods is dev eloped, explored or sustained may be a matter of strate gy and r esource alloc ation, from that of the individual sc ientist to large fundi ng agenc ies. Organising princ iples and con cepts provide a means to facilitate understanding, characterise phenome na, and organise knowledge. Exa mples include emergent scales, tippi ng po ints, phase dia grams, order parameters (collective variables), s ymme try breaking, generalised rigidi ty, n etworks, universality, self-organised criticality, and rugged energy landscapes. Before disc ussing how different characteristics of emergence and methods of inve stigation are ma n ifest i n spe cific sc ientific systems, I discuss the Ising model. 4. Ising models The Ising m odel is emb lematic of toy models tha t ha ve been proposed and st ud ied to understand a nd describe emergent phenomena. Although, originally proposed to descr ibe ferromagneti c phase transitions , variants of it ha v e found application in othe r areas of physics, in biology, economics, sociology, and neuroscience. The ge neral model is defined by a set of lattice points {i}, on each of whic h there is a “spin” {     } and a Hamiltonian where h is the stre ng th of an external magnetic f ield and   is the stre ng th of the interaction betwee n the s pins on lattice sites i a nd j. The simplest models are where the lattice is regular, and the interaction is uniform and onl y non- zero for nearest-neighbour sites. The Ising m odel il lustrates many key feature s of e mergent phenomena. Gi ven the relati v e simpli city of th e model, exhaustive studies over the past one hundred years have given defini tive answers to quest ions that for more com p lex sys tems a re often contentiously debat ed. I enumerate some of these insights: novelty, quantitati v e cha ng e can lead to quali tative change, s pontaneous order, singularities, s hort-range i nteractions can produce long-range orde r, un iversality, mesoscopic scales, s elf -s imilarit y, and that simple models can describe complex behaviour. Most of these propert ies can be illustrated with the case of the Isi ng model on a square lattice with only ne arest-neighbour interactions (     ). Above the critical tempera ture (T c = 2.25J), and in the a bsen ce of a n external magnetic field (h=0) the system has no net magnetisati on. For temperatures bel ow T c , a net magnetisation occ urs,       . F or J > 0 (J < 0) this stat e h as ferromagnetic (anti f erromagnetic) order. Novel ty The state of the system for temperatures below T c is qualit atively different from th e stat e at high er temperatures or from the state of a set of non- inte racting spins. For h=0 the Hamil tonian has the globa l spin-flip symmetry,       However, the low-temperature state does not have this s ymme try. Thus, the non-zero magnetisati on is an emergent prope r ty, as define d by nove lty. Th us, the low-temperature state is associated with spontaneous symmetry breaking , with l ong-r ange order, and with more than one possible equilibrium state, i.e., the magnetisation can be posi tive or negative. Quantitat ive change leads to qualitative change The qua litative change associated with forma tion of the ma gnetic state c an occur with a small quanti tative change in the value of the ratio T/J, i.e., eithe r by d ecrea sing T or inc r easing J, where the range of variation of T/J includes the critical value . Existence of the magnetic state is also associa ted w ith the quantitative c h ange of i n creasing the number of spins from a large finit e number to infinity. For a finite number of spins there is no spontaneous s ymmet ry breaking. Singularit ies For a fi n ite nu mber of spins all the thermodynamic propertie s of th e system a re an analytic function of t h e tempera ture T and magnitude of th e external field h. However, i n the therm odynamic limit, th ermodynamic functions become singul ar at T=T c and h=0. This is the crit ical point in the phase diagram of h versus T. Properties of the whole system, such as the specifi c heat capacity and the magnetic susceptibility, become infinite at the critical point. These singul arities can be characterised by critical exponents , most of which have non- inte ger values. Consequen tly, t h e free energy of the system is not an analytic functi on of the varia bles T and h, over a ny do main containing their critical v alues, T=T c and h=0. Spontaneous order The m agnetic state occurs spontaneously. The system self -organises, even in the presence of therm al fluctuations . There i s no ex ternal field causing the m agnetic state to form. There is long-range orde r, i.e., the v alue of spins th at are infinite ly f ar apart from one another are correlate d. Th is also reflect s a generalised rigidity (t o be discussed later). Short-range interac tions can produce long-range or der. Although in the Ha miltonia n th ere is no direct long-range interaction between spins, long- range order c an occur for t empe r atures less than T c . Prior to Onsager’s exac t solution of the two-dimensional model, publi shed in 1944, many scientists were not convinced that this was possible. 49 Modifi cation of single particle properties In the di sordered state, the probabilit y of a single spin being +1 or -1 is the same, in the absence of a magnetic field. In contrast, in the ordered s tate the probabilities are unequal. Universalit y The va lues of the critical exponents are independent of many details of the model, such as the value of J, the lattice constant and spatial anisotropy, and the presence of small interactions beyond nea rest neighbour. Many details do not matte r. Close to the critical point, the depende nce of sys tem prope rties on the temperature and magnetic f ield is given by uni versal functions. Thi s is why the model can give a quantitat ive de scription of experimental data near the c ritical temperature, even though the model H amiltonian is a crude description of the magne tic in teractions i n a real material. Furthermore, b esides transitions in unia x ial magnets the Ising m odel can also describe transitions and critical behaviour in liquid-gas, binary all oys, and binary liquid mixtures. Structure at the mesos cale There a r e thre e important length scales associated with the model. Two are simple: the lattice constant (whic h d efine s th e spatial sepa ration of neig hbouring spins) , and the size of the whole la ttice. These are the microscopic and macros copic scale, respectively. The third scale is mesoscopic, emerge n t, and temperature de p endent : the correlation length. Th is lengt h is the di stance over which spins are correlated with one anothe r. Th is can also be visualised as the siz e of magnetisation domains se en in Monte Ca r lo simulations, as shown in the Figure below. Fi gure 2. The left, centre , and right panels show a snapshot of a l ikely configuration of the system at a temperature less than, equal to, and greater than the critical temperature, Tc, respect ively. Th e Figure is from Ref. 92 Understanding the connection bet we en the microscopic and macrosc opic properties of the system require s studying the system at the interme diate scale of the correlation length. This scale also defines emergent entities [m agnetic domains] that interact with one another weakly and via an effective interaction. Self-simil arity At the c ritical temperature, the correlation length is infinite. Consequ ently, rescaling the size of the system, a s in a renormalisation group transformation, the state of the systems does not change . Th e system i s said to be scale-free or self-si mila r like a fractal patte rn. This is the inspirat ion for the c on cept of sel f-org anised criticality proposed by Per Bak to e xplain the prevale n ce of power l aws in a wide range of physical, biological, and social systems. 93 Networks of interact ions matt er Propertie s of the system chang e when the network of interacti ons ch anges. This can happen when the topol ogy or d imensi on ality of the lattice changes or when interacti ons beyond nearest neighbours are adde d wi th magnitudes comparable to the nearest-neighbour inte ra cti ons. Th is can cha ng e the relationships bet we en the pa r ts and the whol e. Sometimes deta ils of the parts do matter, eve n produ cing qualitative changes. For example, changing from a t wo-dim ensional r ecta ngu lar lattice to a linear chain t h e ordered state disappears for any non-zero tempe rature. For antiferromagnetic nearest-neighbour interactions changing from a square lattice to a triangular lattice remov es the ordering at finite temperature and lea ds to an infinite number of ground s tat es a t zero temperat ur e. Thus, some microscopic deta ils , such as the network of inte ractions, s ometimes do matter. Knowing the components in a system is not sufficient; knowledg e of how the components i nteract is also required. The m ain point of this example is that to understand a large complex system we need to keep both the parts and the whole in mind. It is not either/or but both/and. Furthermore, there may be an interme d iat e scale, at which new entitie s e merge. Later I briefly discuss how these observati ons are relevant to deb ate s abou t structuralism versus functionalism in biology, psychology, social sciences, and the humanities. I argue that they are trying to defend inte llectual positions (and fashions) th at contradict what the Ising model shows. One cannot preferenc e either the whol e, the parts, or intermediate structure s. Pote ntial and pitfalls of mean-fi eld theory Gi ven its simplicity, s ometimes mean-field works surprisingly well. Other times it f ails specta cularly. This can be illustrated by the Is ing model on a hypercubic lattice of d imension d with nea rest-n eighbour interactions. Mean-field theory predicts a phase t ransition a t a non- zero t emperature for all dimensions. For d=1 t h is is q ualitat ively wrong. For d=2 and 3, this is quali tatively correct but quantitatively incorrect. The critical exponents are different from those give n by mean-fi eld the ory. For d =4 and larger mea n-fi eld theory gives the correct crit ical exponents. Diversity : Si mpl e models can de scribe complex beha viour Consider an Ising m odel w ith competing interactions, i.e. the n eighbouring spins of a parti cular spin compete with one another and wi th an external magneti c field to determine the sign of the spin. An exam p le is an Ising model on a hexagonal close pac k ed (hcp) l attice wi th nearest neighbour antiferrom agnetic interactions and an external magnetic field. The nearest- neighbour i nteractions are frustra ted in the sense t h at if the energy of two neighbouring spins is mini mised by being antiparallel a third adjacent spin must be parallel to one of these spins, frustrating t h e antiferrom agnetic interaction. Th e hcp lattice can be viewed as layers of hexagona l (triangular) lattices w here each layer i s di splaced relative t o ad jacent layers. This mode l has been studied by materials scientists as it can describe the ma ny possible phases of binary a lloys, A x B 1-x , where A and B are different chemica l elements (for example, silver a nd gold) and the Ising spins on site i ha s va lue +1 or -1, corresponding t o the presence of atom A or B on t hat site. The magnetic field corresponds to the difference i n the chemical potent ials of A and B and so is related to thei r r elative conce n tration. 94 A study of this model 95 on the hexagonal close packed lattice at zero tem perature found r ich phase diagram s incl uding 32 st able ground st ates with stoichiometries , includi ng A, AB, A 2 B, A 3 B, A 3 B 2 , and A 4 B 3 . Of these struct ures, six are stabilized by p urely nea r est-neighbour i n teractions, eight by add iti on of n ext-nearest neighbour interactions . With multiplet i n teractions, 18 more distinc t structures become stable. Ev en for a single stoichiometry, there ca n b e multiple possible disti nct orderings (and crystal structures). A second exam ple is the Anisotrop ic Next-Nearest Neighbour Ising (ANNNI) model on a cubic lattice. I t supports a plet hor a of ordered state s , even t hough the model has only two paramete rs. A graph of the wave v ector of the ground state versus the anisotropy parameter or tem perature h as a fractal structure, and is known as the Devil’s stai rcase. 96,97 These t wo Is ing models ill ustr ate how relati v ely simple m odels , containing competing inte ra cti ons (descr ibed by just a few parameters) can describe rich behaviour, p articularly a diversit y of ground sta tes. They also illustrate how s ometimes networks of interactions do mat ter, leading to qualitatively different behaviour. Dynamical f ormulations The ori ginal formulation of the Ising model has no dynamics and the focus was on properties in the rmodynamic equilibrium. How ever, i t can also be formulated with simple discrete dynami cs and sto chastic proce sses. These dynamical models can describe non-equilibrium phenome na such as metastability, hysteresis , domain nucleation, a nd coarsening. Th is formula tion has been fruitful in applying the model to problems in fi n ance and economics, as discussed lat er. One v ersion, known as Gla ub er dynamics, is defi n ed by   󰇛  󰇜        󰇛    󰇜       󰇛  󰇜  where   󰇛  󰇜 is randoml y d istributed. This equation belongs to the class of stochastic dynami cal models of interacting particles which have been much studied mathematic ally and exte nsively by Ligg e tt. 98 Cross -discipl inarity Beyond ma gnetism, Ising-l ike models are toy models in othe r fields, some of which will be discussed below. Exa mples include Z 2 lattice gauge theory, the Edwards-Anderson model for spin glasses, the Hopfiel d n eural network model for associat ive memory, Hinton’s Boltzmann mac hine for artificial intelligence, th e random fiel d I sing model i n economics, and the Eaton- Munoz model for protein folding. 5. Stratific ation of ph ysics Scale s, s trata, and subdisciplines The re is a str atum of sub-disc iplines of physics, illustrated in Table 1. For ea ch stratum, there are a ra nge of length, time, and energy scales that are relevant. There are distinct entities t h at are com posed of the entities from lower stra ta. T h ese composite entities interact with one anothe r via effective interactions that arise due to the inte ractions present at lower strata and can be described by an effective theory. Each sub-discipli n e of physics is semi-autonomous. Colle ctive phenomena ass ociated with a si ng le strat um can be studied, described, and understood without re f erence to lower strat a. Tabl e entries are not meant to be exh austive bu t to illustrate how emergence is central to understandi ng sub-fie lds of physics and how they a re related and not related to ot h er sub- fiel ds. S ub-disciplines ca n also be defi n ed sociol og ically. For each sub-discipline t h ere are depart ments, courses, tex tbooks, professional organisations, confe rences, journals, and funding programs. Sub - field of physics Scale, length (metres) Scale, energy (eV) Entities Effective interactions Effective Theory Collective Phenomena Cosmology 10 21 - 10 26 Cosmic microwave background, galaxies, dark matter, dark energy Curvature of space - time Equation of state, General relativity Uniformity and isotropy Stellar 10 8 - 10 12 Protons, neutrons, helium, electrons, neutrinos Buoyancy, Radiation pressure, Fermi degeneracy pressure Equation of state Neutron stars, white dwarfs, supernovae, Earth science 10 3 - 10 8 Clouds, oceans, currents, tectonic plates Climate models Earthquakes, tornadoes, global warming Fluid dynamics 10 -3 - 10 3 Vortices, Rayleigh - Benard cells , Boundary layers, Thermal plumes Buoyancy, convection, Kolmogorov Navier - Stokes equation , mixing length model Turbulence, pattern formation Condensed Matter 10 -10 - 10 - 2 10 -4 - 10 Electrons, nuclei, Quasiparticles Screened Coulomb, pseudopotentials, electron -phonon, BCS attraction Ginzburg - Landau, Fermi liquid States of matter, Topological order, Spontaneous Symmetry Breaking Molecular 10 -10 - 10 - 8 10 -3 - 10 Molecules van der Waals, H-bonds, ionic, and covalent Lennard - Jones, Potential energy surface Chemical bonding and reactions Atomic 10 -11 - 10 -9 1-100 Atoms Hartree - Fock Periodic table Nuclear 10 -15 10 3 - 10 8 Neutrons, protons, nuclei Yukawa, Nuclear mean - field Shell model , Interacting Boson, liquid drop Magic numbers , fission, fusion, beta-decay Elementary particles and fields 10 -15 - 10 - 15 10 6 - 10 12 Quarks, l eptons, photons, g luons, W bosons, Higgs boson Gluon exchange Standard model Mesons, Hadrons, Spontaneous Symmetry Breaking, Quark confinement, Asymptotic freedom Early universe 10 -8 - 10 12 10 24 -10 28 Topological defects, Inflatons GUTs Anti - matter annihilation, phase transitions , inflation Quantum gravity 10 -35 10 28 Gravitons Hawking radiation, Black hole evaporation Tabl e 1. Sub-disc ipli n es of physics and the a ssociated s cal es, effective interactions and theori es, and ph enomena. Ti me scales generally decr ease from t h e top row s to the bottom rows of the Table . Someti mes s imilar physic s occurs at vastly different scales. For exam p le, Bay m discusses simil arities of the physics associated with cold atomic gases and qua rk -gluon p lasmas. 99 These sim ilarities occur in spite of the fact that the relevant energy scales in t he two s ystems differ by more than 20 orders of m agnitude. 6. Classical physics 6.1 Theory of elasticity This is an e ffective theory that describes smooth small distortions of solids over l ength scales much l arger than atomic scales. It can be used to illustrate several aspects of em ergence, parti cularly relating to the connection between macro - and micro- sc al es. Novel ty. El asticity (ri g idity) is an eme rg ent property of a solid. The atoms that make up the solid do not ha ve this property. Shear rigidity is a property that liquids do not have. Elasticity gives rise t o the propagation of sound through a crystal. Intra-stratum closure . The the ory g ives a complete descripti on of ph enomena on long le ng th scale s. There is no need for information from shorter length scales or microscopic details. Historica lly, the theory was completed before microscopic details such as crystal structures o r quantum theory were known. . Universalit y . The theory is inde pendent of the chemi cal composition of the solid or the origin of physical forces between the constituent atoms. The same elasticit y theory i s the continuum lim it of both classical and quantum theories for the interactions and dynamics of the constit uent atoms. Laughlin and Pines emphasised that this universality of the theory hides the unde rlying microscopic physics, calling it a protectorate. 61 Energy functiona l has a simple form, being quadratic in gradients of the local de v iations of the soli d from uniformity. For a cubic crystal there a re three independent elastic constants, C 11 , C 12 , and C 44 . The latter determi nes the shear m odulus. Macro- properties can provi de hints of the micro- pr opertie s. Marder g ave two e xamples. 100 [The fi rst is on p.290 and the second on p. 305 in his first edition]. Fi rst, relationships between the elastic constants constrain the n ature of mic roscopic forces. Cauchy a nd Saint Venant showed that if all the atoms in a crysta l interact through pair -wise cent ra l force s then C 44 =C 12 . However, in a wide r ange of elemental crystals, one finds that C 12 is one to t hree ti mes la rg er than C 44 . This discrepancy caused significant debate i n the 19th ce ntury but was resolved in 1914 by Born who showed that a ngular forces between atom s could explain the violation of this identity. Fr o m a qua ntum chemical perspective, these a ngular forces arise because it costs energy to bend chemical bonds. Second, eve n before the discovery of x-ray diffr action by crystals, there were strong reasons to bel ieve in the microscopic periodic structure of crystals. The first paper on the dynamics of a crysta l lattice was by Born and von Karman in 1912. This prece d ed the famous x -ray diffraction e xperiment of von Laue that established the underlying crystal lattice . In 1965, Born had the following reflect ion. The fi rst paper by Karman and myself was published before La u e's discove ry. We regarded the e xistence of lattices as evident not only because we knew the group theory of lattic es as given by Schoe nflies and Fedorov whic h explained the geometrical features of crystals, but also bec ause a short time before Erwin Madelung in G  ttingen had derived the first dynami cal inference from lattice theory, a relation between the infra -red vibration frequency of a crysta l and its elastic properties.... Von Laue' s paper on X -ray diffraction whic h g ave direct e v idence of the lattice stru cture appea red b etwee n our first and second paper. Now it is remarka b le that in our se cond paper there i s also no reference to von Laue. I can explain this only by assuming tha t the concept of the l attice seemed to us so well established that we regarded von Laue' s work as a welcome confirm ation but not as a new and exciting discovery which it reall y was. Distinct ly different types of sound, distinguished by their spee d and their polari sation, c an travel through a pa r ticular crystal are macrosc op ic manifestations of the spe cific translational and rota tional symmetries that the crystal has at the microscopic le v el. T he highest sym met ry crystal is a cubic crystal. It has three distinct types of sound. In a crystal with no rotational or mirror symm etry there are many more d ifferent sound types. Hence, measurement of all the different types of sound in a crystal can provide informati on about the symmetry of the crystal . The t heory only describes s mall deviations from uniformity and so cannot describe objects that occur at the mesoscale, such as disinclinations and dislocations, tha t are topological defect s. A hint of this failure was provided in the 1920s when it was found that the observed hardness of real materials was orders of magnitude s mal ler than predicted by the theory. 6.2 Thermodynamics and Brownian motion Thermodynamics Novel ty . Te mperat ur e and entropy are emergent properties. Classically, they are defined by the z eroth and second law of thermodynamics, respectively. The individual particles that make up a system in thermodynamic equilibrium do not have these properties. Kadanoff gave an exa mple of how there is a qualitative difference in macro- and micro- pe rspe ctives. He pointe d out how deterministic behaviour can emerge at the mac rosc ale from sto chastic behavi our a t the microscale. 101 The many individual molecules in a dilute gas can be v iewed as undergoing stoc hastic motion. However, collectively they are descri b ed by an equation of state such as the ideal gas law. Prim as gave a technical argument, involving C* algebras, that tem perature is emerge n t: it belongs to an algebra of c ontextual observables but not to the alge bra of intrinsic observable s. 54 Fol lowing this per spective, Bishop argues that temperat ur e and the c h emical po tential a r e (contextually) emergent. 11 Tem perature is a macroscopic entity that is defined by the ze ro th law of thermodynamic s. Intra-stratum closure. The la ws of ther modynamics, the equations of the r modynamics (such as TdS = dU + pdV), and state functions such as S(U,V), provide a c omplete description of processes involving e qu ilibrium states. A knowledge of microscopic detail s such as the atom ic constituents or forces of interaction is not necess ary for the de scription. Irreducibil ity . A com mon vie w is th at thermodynamics can be de r ived from statistica l mec hanics. The ph ilosopher of sci ence, Ernst Nag el claimed this was an example of t heory reducti on. 102 However, this is contentious. David De utsch claim ed that the second law of therm odynamics is an “emergent law”: it cannot be derived from microscopic la ws, like the princi ple of testability. 103 Lieb and Yngvason stated that the derivation from statistical mec hanics of the law of entropy increase “ is a goal that has so far eluded the deepest thinke rs. ” 104 In contrast, Weinberg claimed that Ma xwell, Boltzmann, and Gibbs “showed that the principles of thermodynamics could in fact be deduced mathematically, by an anal ysis of the prob abilities of different configurations… N everthe less, even though therm odynamics has been explained in terms of particles and forces, it continues to deal wit h eme rgen t conc epts like temperature and entropy that lose all meaning on the level of indivi dual particles.” 105 (pages 40-41) I agree that thermodynamic properties (e.g., equations of state , and the temperature dependence of heat capacity, and phase t r ansitions) ca n b e deduced from st atistical me chanics. However, th ermodynamic principles , such as the second law, are not thermodynamic prop erties. These thermodynamic principles are re qu ired to justify t he equations of s tati stical mechanics, such as the partition function, that are used to cal culate thermodynamic properties. Thus, cla iming that temperature is an emergent entity and irreducible may be related to t h e question a s to whether the zeroth law can be derived from mic roscop ic laws. Earl ier, I mentioned how macroscopic thermodynamics provided hints of underlying mic roscopic phys ics through the Third law, Gibbs paradox, and the Sackur-Tet rod e equation. Brownian motion Novel ty. The r andom beha v iour of Brownian motion emerges from underlying deterministic dynami cs a t the microscale. 101 The diffusion equation breaks time-reversal symmetry whereas the underlying micros copic dyn amics has time-reversal invariance. Sethna di scussed how Brownian motion illustrates emergence. 106 Consider a large num b er of steps of the same size in random directions. Th e associa ted random walks have a scale invari ant (frac tal or sel f-si milar ) structure in time and space. i.e., if the time is rescaled by a factor N and t h e le ng th is rescaled by a factor   , then the walk looks the same. Secondly, the dista n ce from the original position is gi ven by a probabil ity distribution that is a solution to the diffusion equation. Both proper ties are un iversal , i.e., they are independent of most deta ils of the random walk and so apply to a wide range of sys tems in physics, chemistry, and biology. 6.4 Fluid dynamics and p attern formation Wea ther involves many scales of distance, time, and energy. Describi ng weather means maki ng decisions about what range of scales to focus on. Key physics involves thermal convec tion which reflects an interplay of gravity, the rmal expansion, viscosity and t her mal conduct ion. This can lead to Rayleigh-Bénard convection a nd convection cells. Flui d dynami cs Novel ty. Turbul ent flow i s qua litatively diffe r ent from normal un iform flow. In th e transition to turbul ent hea t flow meso-scale struct ur es emerge, such as plumes, flywheels, jets, and boundary la yers. Discontinui ties. Ther e are qualitati ve changes in fluid flow w hen a dimensionless parameter such as Raylei gh nu mber or Reynolds number passes through some c ritical value. Kadanoff descri bes three levels of description for co nvecti v e turbulence, including laws unique t o each level. 107 Multiple scales are associated with multiple entities: -the molec ules that make up the fluid -small volumes of fluid that are in local thermodyna mic equilibrium with a well -defined tem perature, density, and velocity -individual c onvection cells (rolls) -collect ions of cells. At eac h scale, the corresponding entities can be viewed as emerging from the inte r acting enti ties at the next smallest scale. Hence, at each scale the entities are collective degrees of freedom. Unpredictabi lity. In pr inciple, a c o mplete de scr iption, includi ng the t r ansition to turbulence, is given by t he equations of fluid dynamics, including the Navier-Stok es equation. Despite the a pparent simplicity of these equations, making definitive pre d ictions from them rema ins elusive . A toy model is the lattice gas on a hexagonal lattice. It can describe the complex structures seen for inc ompress ible fl u id flow behi nd a cylinder for a Reynolds number of about 300. 108 Universalit y. Si milar properties occ ur for flu ids with diverse chemical com posi tion and origins of the molecular forces responsible for fluid properties such as vi scosi ty or thermal conduct ivity. E mergent properties often depend on dimensionless parameters that are combi nations of several physical parameters, such as the Reynolds number or the dime nsionless heat fl ow Nu. There are universal scal ing la ws, s uch as relating dimensi onless heat flow to temperature difference or th e probability distributi on for temperature fluct u ations in a c ell. 107 For t urbulence, Kol mogorov proposed a scaling law for the probability distribut ion of the scales at which energy i s tr ansferred from larger eddies to smaller ones without signi ficant viscous dissipation. 109 Singularit ies . Consider the incompressible Euler equations in the limit that the vi scosi ty tends to ze ro. Onsager argued that there is non-zero energy diss ipati on in this limit and tha t the veloc ity field does not remain differentiable. 110 – 112 This singularity is central to Kolmogorov’s theory of t urbulence. Another cla ss of s ingula rities occurs wh en a mass of fluid forms a t h in nec k and that neck can breaks so that the fluid breaks into two pieces. There a r e the n singu larities i n the mathematical solutions to the fluid dynamic equations. 113 This is al so relevant to fluid flow on a surface that is partially wet and dry. 114 Patt ern formation Patte rns in space and/or time fo rm in fl u id dynamics (Raylei gh-B ernard convecti on and Tayl or-Coue tte flow), laser physics, materials scie n ce (dendrites in forma tion of soli ds from liqui d melts), bio logy (morphogenesis), and chemistry (Belousov-Zhabotinsky reactions). Most of these systems are driven out of equilibrium by external constraints such as tem perature gradients. Novel ty. The p arts of the system can be viewed as the molecular constituents or small uniform pa rts of the sys tem. In eithe r case, the whole s ystem has a prope rty ( a pattern) that the pa rts do not have. Discontinui ty. The system makes a transition from a uniform state to a non-un iform state when some para meter becomes larger than a critical value. Universalit y. Si milar patterns such as conve ction roll s in fluids can be observed in diverse systems regardle ss of the microscopic details of the f luid. Often there is a single parameter, such as the Re ynolds number, which involves a combinati on of f luid properties, that dete rmines the type of patterns that form. Cross and Hohenberg 115 highlight ed how the model s and mechanisms of pattern formation across phys ics, chem istry, and biology have simil arities. Turing’s model for pattern form ation in biology associated it with concentration gradient s of re acting and di ffusing molecules. However, Gierer and Mei nh ardt 116 showed that all that was required was a network with competition betwee n short-range pos itive feedback and long-range n egative fee db ack. This could occur in circuit of cellular signals. This ill ustrates the problem of protectorates. Self-organisati on. The formation of a particul ar patte rn occurs spontaneousl y, resul ting from the i nteraction of the many components of the system . Eff ective theories. A crystal growing from a li qu id me lt c an form shapes such a s dendrites. This proce ss involves instabilities of the s hape of the crystal-liquid interface. The interface dynami cs ar e completely desc ribed by a few partial differential equati ons that can be derived from mac roscop ic laws of thermodynamics and heat conduction. 117 Diversity . Diverse p atterns are observe d, particularly in biological systems. In toy models, such as the Turi ng model, with just a few paramete rs , a diverse range of patterns, both in time and space , can be produced by varying th e parameters. Many repeate d iterations ca n lead to a diversit y of struc tures. This may result from a sensitive depende n ce on initial conditi ons and history. For e x ample, each snowflake is different bec ause as it falls it passes through a slightl y different environment, with sm all va r iations in te mperature and humidity, co mpared to others. 118 Toy models. Turing proposed a model for morphogenesis in 1952 tha t involved two coupled reaction-diffusion e quations. H omogene ous con centrations of the two chemicals become unstable when the difference between the two diffusion constants becomes sufficientl y large. A two-dimensional v ersion of the model can produce diverse pa tterns, many resembling those found in a nimals. 119 However, after more than s eventy yea rs of ex tensive study, many devel opmental biologists remain s ceptical of the relevance of the model, partly because it is not cl ear whether it has a microscopic basis. K icheva et al., 120 argued th at “patte rn for mation is an emergent behaviour that result s from the coordination of ev ents occurring across mole cular, cellular, and tiss ue scale s.” Oth er toy models include Diffusion Limited Aggregation, due to Witte n and Sander, 121,122 and Barnsley ’s iterated function system for fractals tha t produces a pattern like a fern. 123 6.6 Caustics in op tics Michae l Berry highl ighted how ca ust ics illustrate emergenc e. 55,124,125 He stated that “ A causti c is a collective phenomenon, a prop erty of a fami ly of rays that is not present in any indivi dual ray. Probably the m ost f amiliar e x ample is the rainbow. ” Caustic s are envelopes of families of rays on which the inte nsi ty diverges. They occ ur in medi a where the refractive index is inhomogeneous. For example, the cells seen in bodies of sunlit wat er occur due to an interplay of the uneven a ir-water interface and t h e difference in the refracti ve index between air and water. For rainbow s, key param eters are the refractive index of t he water droplets and the size of the droplets. The ca ust ic is not the "rainbow", i. e., the spec trum of colours, but rather the large ligh t intensity associ ated with the bow. Caustic s illustrate several characteristics of emergen t properti es: novelty, singularities, hierarc h ies, new scal es, effective theories, and univers alit y. Novel ty. The whol e system (a family of light rays) has a property (infinity intensity) that indivi dual light rays do not. Discontinui ties. A caustic defi n es a spatial boundary across which the re are discontinuities in propertie s. Irreducibil ity and singular limits. Causti cs only occur in the theory of geometrical optics which corre sponds to the limit where the wavelength of light goes to zero in a wave the ory of light . Caustics (singularities) are not present in the wave the ory. Hierarchies. These occur in t wo different ways. First , light can be treated at the level of rays, scala r waves, and vector waves. At each level, there are qualitatively diffe rent singularities: causti cs, phase singularities (vortices, wavefront dislocations, nodal lines), and polarisation singulari ties. S econd, treating caustics at the level of wave the ory, as pioneered by George Bidel l Airy, reveals a hierarchy of non-ana lyticities, and an interference pattern, re f lected in the supernum erary part of a rainbow. New (emergent ) scales. An example is the universal angle of 42 de gr ees subtended by t h e rainbow, tha t was first calculated by Rene Descartes. Airy's wave theory showed t h at the spacing of t he interference fringes shrinks as   , where  is the wavelength. Eff ective theories. A t eac h level of the h ierarchy, one can define and inve s tigate effecti v e theori es. For ray t h eory, the e ff ective t h eory is defined by the spatia lly dependent refractive index a nd the ray action. Universalit y. C austic s ex ist for any kind of waves: light, sound, and matter. They exhibit "structural sta b ility". They fa ll into equivalence (uni versali ty) classes that are defined by the ele mentary catastrophes enumerated by Rene Thom and Vladimir Arnold . 124 Any two mem bers of a class can be smoothly deformed into one another. K is the number of paramete rs needed to define the class and the associated polynomial. 6.7 Chaos th eory The ori gins of the field lie with two toy models: Lorenz’s model and the logistic iterative map. In 1975, Robe rt May published an influential review e ntitled, “ Simple mathematical model s with very complicated dynamics .” 126 I now r eview the two models a nd how th ey ill ustrate how em ergent prope r ties occur in system s i n which there are many iterations. Lorenz model This model was studied by the meteorologist Edward Lorenz in 1963, in a seminal paper, "Determi nistic Nonperiodic Flow." 127 Under the re str icti v e conditions of conside r ing the dynami cs of a single convection roll the model can be derive d fro m the full hydrodynamic equat ions descr ibing Ra y leigh-Bernard convection. Lorenz's work and its impact is beaut ifully described in James Gleick's book Chaos : The Making of a Ne w S cie n ce . 82 The m odel consists of just three coupled ordinary differential equations: The va riables x(t), y(t), and z(t) describe, respectively, the am p litude of the velocity mode , the t emperature mode, and the mode measuring the h eat flux Nu, the Nus selt numbe r. x and y characte rize the roll pattern. The model has t hr ee dimensionle ss para meters: r,  , and b. r is the ra tio of the temperature difference between the hot and cold plate, to its critic al value for the onset of conv ection. It c an also be viewed as the ratio of the Rayleigh number to its crit ical value.  is the Prandtl number, the ratio of th e kinemat ic viscosity to the thermal diffusivit y.  is abou t 0.7 in air and 7 i n wa ter. Lore n z used  = 10. b is of order unity and convent ionally taken to have the value 8/3. It arises from the nonlinear coupling of the fluid veloc ity and temperature gradient in the Boussinesq approximation to the full hydrodynamic equat ions. The m odel is a toy model because for values of r larger than r c (defined below) the approxim ation of three modes for th e full hydrodyn amic equations describing thermal convec tion is not physic ally realistic. Neve r theless, mathematically the model s hows its most fascina ting properties in that parameter regime. Novel ty The m odel has several distinct types of long- time dynamics: stable fixed points (no convec tion), limit cycles (convective rolls), and most strikingly a chaotic strange att r act or (represente d below). The chaos is reflected in the sensitive de p endence on ini tial conditions. Briefl y, a strange attractor is a curve of infinite length that ne v er crosses itself a nd is conta ined in a finite volume. This means it has a fractal structure and a non -trivial Hausdorff dime nsion, c alc u lated by Viswanath 128 to be 2.0627160. Discontinui ties Quantit ative changes lead to qualitative changes. For r < 1, no convection occurs. For r > 1, convec tive rolls develop, but these become unstable for and a stra nge attractor develops. Phase diagram Lorenz onl y considere d one set of parameter values [r =28,  =10, and b=8/3]. This was rather fortunate , b eca use then the str ange attractor was waiting to be discovered. The phase diagram shown below maps 129 out the qualitatively different behaviours that occur as a functi on of sigma (ve rtical axis) and r (horizontal axis). Different pha ses are the fixed points P± associated with convective rolls (black), orbit s of period 2 (red), pe r iod 4 (green), pe r iod 8 (blue ), and chaotic attractors (white). Universalit y The de tails of the molecular composition of the fluid and the intermolecular interactions a re irrele v ant be yond how th ey dete r mine the three parameters in the model. Hence , qu alitativel y simil ar behaviour can occur in systems with a wide range of chemical compositions and physical properties. Th e strange attractor also occurs for a wide range of pa r ameters in t h e model. Unpredictabi lity Although the system of three ord inary diffe r ential equations is simple, discove ry of the strange a ttractor and the chaotic dynamics was unanticipated. Furthe r more, the dyn amics in the c haotic regime are unpredictable, given the sensitivity to initial conditions. Top-down causation The propertie s and behaviour of the system are not jus t det ermined by the properties of the mole cules and their interactions. The external boundary conditions, the applied tempe r ature gradient and the spa tial separation of t he hot and col d plates, are just as i mportant in dete rmining the dynamics of the system, including motion as much smaller length scal es. Logistic i terative map The i terative map w as studied by Feige nb aum who discovered universal properties that he expla ined using renormalisation group techniques de veloped t o d escribe critical phenome na. 130 The state of t he system is defined by infinite discrete “time series” 󰇝            󰇞 . The “i n teraction” b etwee n the    is defined by the iterative map      󰇛     󰇜 where the parameter  can be vie wed as the stre ng th of the interactions. Novel ty. Ordered state s are p eriodic orbits to which the system converges regardle ss of the initial state x 0 . These sta tes are qualitatively different from on e another, just like different crystals with different sym met ry are different from one another. The chaotic orbits are qualitatively different fro m the periodic orbits. Discontinui t ies, ti pp ing points, and phase diagram. Quantitative change in  leads to qualitati v e cha ng e i n behaviour. As  increases “ period doubling ” occurs. At     , there is a tra nsition from and orbit with pe riod n to one with period 2n. F or      the dynamics is chaotic, i.e., there is no periodicity and there is sensitivi ty to the initial value, x 0 . For larger  , periodicit y r eappears for some finite range of  , before chaotic be haviour recurs. The associated bifurcati on d iagram also exhi b its self- simil arity. Universalit y. Fe igenbaum nu merically observed t he sc aling relationship         where  = 4.6692... is the fi rst Feigenba u m constant. He conjectured this was true for any ite ra tive map, x n +1 = f(x n ) where f(x) is of s imil ar shape to the logistic map, and gave a justifi cation inspired by the renormalisation group. T his conjecture was rigorously proven by Colle t, Eckmann, and Lanford. 131 They also showed how logistic map c ould describe transit ion to turbulence. Libcha b er performed experi ment s s howing the period doubling transit ion to turbul ence in Rayleigh-Bernard convection cells, obs erving a value of        . 132 6.8 Sychronisation of oscillator s When a large num ber of harmonic oscillators with di fferent fre quencies inter act with one anothe r they can undergo a transition to a col lective state i n whi ch they all oscillate with the same fre quency. This phenomenon has bee n se en in a diverse range of systems inc luding arrays of Josephson juncti ons , neural networks, fireflies , and London’s Millenium bridge. 133,134 Kuramoto proposed a t oy model 135 that can describe the essential aspect of the ph enomena. It describe s a phase transition that o ccurs at a critical value of the coupling strength. For weak coupli ng the oscillators are incohe r ent, oscillating at different frequencies. Above the c r itical coupli ng they os cillate in pha se at the same frequenc y. 6.9 Arrow of ti m e Time h as a direction. Microscopic equations of motion in classical a nd qu antum mecha n ics have t ime-rev ersible sym metry. But this symmetry is broken for many macroscopic phenome na. This observation is enc oded in the seco nd law of therm odyn amics. We experi ence the flow of time and distinguish past, present, and future . The arrow of time is mani fest in phenomena that occur at scales cove r ing many orders of magnitude. B elow, six different arrows of time a r e listed in order of increasing time scales. Th e relati onship be tween some of the m is unclear. These are discussed by Leg gett i n chapter 5 of The Problems of Physic s 136 and more rece n tly reviewed by Ellis and Dros sel. 137 Ele mentary particle physics. CP viol ation is observed in certain phe no mena a ssociated with the wea k nuclear interaction, such as the decay of neutral ka ons obs erved in 1964. The CPT symmet ry th eorem shows that any loc al quantum field theory that is invariant under the “proper” L orentz transformations must also be invariant under co mbine d CPT transforma tions. This me ans that CP violation means that t ime-reversal symmetry i s broken. In 1989, the di rect violation of T symmetry was observed . Ele ctrom agnetism. When an electric charge is accelerate d an electromagnetic wave propagate s out fro m the charge towards infinity. Energy is transferred from the charge to its environm ent. We do no t observe a wa ve that propag ates from infinit y into t h e accel erating charge, i . e., energy being tra nsferr ed from the envi ro nment t o the charge. Yet this pos sibility is all owed by the equations of motion for electromagnetism. There is an absence of the “adva nced” solut ion to the equations of motion. Thermodynamics. Irreversibility occurs in i so lated systems. They tend towards equilibrium and uniform ity. H eat never tra v els from a cold body to a hotter one. Flui ds spontan eously mix. The r e is a time ord ering of the therm odyn amic s tat es of isolated macroscopic systems. The t hermodynamic entropy encodes this ordering. Psychol ogical experience. We remember the past an d we think that we can affect the fut ur e. We don’t think we can affect the past or know th e future. Biol ogical evolution. O ver time spec ies adapt to their environment and become more compl ex and more diverse. Cosmology. There was a be g inning t o the unive rse. The universe is expanding not contracti ng. Density p erturba tions grow independe n t of cosmic time (Hawking and Lafl amme). The probl em of how s tati st ical me chanics connects ti me-reversible microscopi c dynamics with ma croscopic irr eversibility i s subt le and c on tentious. Lebowitz claim ed that this problem was solved by Boltz mann, provided that the distinction between typical and average behavi our ar e accept ed, along with the Past Hypothesis. 138 This was advoca ted by Albert 139 and state s th at the universe was initi ally in a state of ext remely low entropy. Wallace discussed the nee d to accept t h e idea of probabilities in law of physics and that th e competi ng inte rpre tations of probability a s frequency or ignorance matter. The Past Hypothe sis is fascin ating because i t connects the arrow of time seen in the laborat ory and everyda y life (time scales of microsec onds to years) to cosmol ogy, covering tim escales of the lif etime of the un iverse (10 10 years) and the “initial” state of the universe, perhaps at the end of the inflationary epoch (10 -33 seconds ). Th is also raises questions about how to formula te the Se cond Law and the concept of entropy in the presence of gr avity and on cosmologi cal l ength and time scale s. 140,141 7. Quantum-classical boundary Classica l physics emerges from quantum physics in the limit that Planck ’s constant tends to zero. Berry poi n ts out that thi s l imit is a singular asy mptotic expa nsion. 55 Quantum measurement problem One of the bi ggest challenges in the foundations of physics is the quant u m measurement problem . It is ass ociated wit h a few key (distinct but related) questions. i. How does a mea surement convert a coherent state undergoing unitary dynamics to a "classica l" mixed state for which we can talk about probabilitie s of out comes? ii. W hy is the outcome of an individual measurement always defi nite for the "pointer states" of the m easuring apparatus? iii . Can one derive the Born rule, which gives the probability of a particular outc o me? Emergence of the class ical world from the quantum wor ld vi a decoherence A quantum system always interacts to some extent with it s env ironme n t. This interaction lea ds to decoherence, whereby quantum interference effects are washed ou t. 142 Consequently, superposition sta tes of the system decay into mixed s tate s described by a diagonal density mat rix. A major research goal of the past three decades has been understa nd ing decohe r ence and the extent to which it does provide answers to the quantum measurement problem. One achi evement is that decoherence theory seems to give a mechanism and time sca le for the “col lapse of the wavefunction” within the framework of unitary dynamics. However, this is not the case because decoherence is not the same as a projection (which is what a singl e quantum measurement is). Decoherence does not produce definite outcomes but rather stati stical mixtures. D ecohe rence only resolves the iss ue if one i dentifies ensembles of mea sured states with ensembles of the decoher ed density matrix (the stati st ical interpretati on of quantum mec hanics). Thus, it seems decoherence only answers the first que st ion above , but not t he last two. On the other hand, Zurek has pu shed the de coherence picture further and given a “derivation” o f the Born rule within its framew ork, with an additional assumption that he dubs quan tum Darwini sm. 143 In other words, decoherence does not solve the quantum mea surement problem: measurements always produce definite outcome s. One approac h to solving the qu antum measurement problem is to view quantum theory as only an a pproximate theory. It could be an effective theory for some underlying theory valid at t ime and length scales much smaller than those for w hich quant um theory has been precise ly tested by experiments. Emergence of quantum theory from a “classical” statistical theory Einste in did not accept the statistical nature of quantum theory and considered it should be deriva ble from a more “realistic” theory. In particular, he suggested “a complete physical descript ion, the statistical quantum theory would …. take an approximately analogous position t o the statistical mechanics within the framew ork of classic al mechanics. ” 144 Einste in's challenge was taken up in a concrete and impressive fashion by Stephen Adler i n a book, “Quantum Theory as an Emergent Phenomenon: The Statistical Mechanics of Matrix Models as the Prec ursor of Quantum Field Theory”, 145 published i n 2004. A helpful summary i s given in a review by Pearle. 146 The sta rting point is " classica l" dynamical variables qr and pr which are NxN mat r ices, where N is even. Half of t hese variables are bosonic, and the others are fe r mionic. They all obey Hamil ton's equations of motion for an unspecified H ami ltonian H. Three quantities are conserved: H, the fermion number N, and (very imp ortantl y) the traceless anti -self-adjoint mat rix, where the first term is the sum for all the bosonic variables of their commutator, and t h e second is the sum over anti- commutators for the f ermionic variables. Quantum t heory is obtained by tracing over all the classical va r iables with respect to a canoni cal ensemble with three (matrix) Lagrange multipliers [analogues of tempera ture and chem ical potential in conventional statistical mechanics] corresponding to th e conserved quanti ties H, N , and C. The e xp ect ation values of the diagonal elements of C are assumed to all have the same value, hbar! An analogy of t he equipartition theorem in classical s tati stical mechanics (which looks like a Ward i dentity in quantum field theory) leads to dynamical equations (tra ce dynamics) for effective fields. To make these equations look like regular quantum fiel d theory, an assumption i s made about a hierarchy of length, energy, and "temperature" [La gr ange mult iplier] scales, which cause the Trace dynamics to be dominated by C rather tha n H, the trace Hamiltonian. Adler suggests these scales may be Planck scales. Then, the usual quantum dynamical equations and the Dirac corresp ondence of Poisson brackets and comm utators emerge. Most of the actual details of the trace Hamiltoni an H do not matter; anothe r c ase of unive rsality, a common characteristic of emergent phenom ena. The “ classical” field C fluctuates about its average value. These fluctuati ons c an be identified with corre ctions to locality in quantum field theory and with the noise terms which appear i n the m odified Schrodinger equation of "physical colla pse" models of quantum theory . 147 , 148 More recently, t heorists including Gerard t’Hooft and John Preskill have i nv estigated how quantum mechanics can emerge from other deterministic systems. This is sometimes known as the e mergent quantum mechanics (EmQM) hypothesis. Underlying deterministic systems considere d include Ha milton-Randers systems defi n ed in co-tangent spaces of large dime nsional configuration spaces, 149 neural networks, 150 cel lular automata, 151 fast moving cla ssical variables, 152 and the boundary of a local clas sical model with a length that is exponent ially large in the number of qbuits in the quantum system. 153 The fact t h at quant u m theory ca n eme rge from such a diverse range of unde rlying the or ies again illustrates universal ity. In most of th ese versions of Em QM the length scale at which the und erlyi ng theory bec o mes rel evant is conjectured to be of the order of the Planck length. The que stion of quantum physics emerging from an underlying classical theory is not just a question i n the foundations of phys ics or in phil osop hy. Slagle poi n ts out 154 that Emergent Quantum Mec hanics may mean that the computational power of quantum computers is severel y limited. He has proposed a specific experimental protocol to test for EmQM. A large number d of entangl ing gates (the circuit depth d) is applied to n qbits in the computational basis, followed by t he inverse gates. This is followed by measurement i n the c o mputational basis. The fi delity should decay exponentially with d, w hereas for Em QM will decay much faster above some critical d, for su fficiently large n. Slagle and Preski ll justified their work on EmQM as follows. 153 “ A characte ristic signature of emergence at low energy or long time and distance scale s is that the resulting physics is typically well described by remarkabl y si mple equat ions, which are often linear (e.g., the harmonic os cil lator) or only consist of lowest order te r ms in an e ffective Lagrangian. ” In principle, i t is pos sible that quant um mechanics is also only an approximate descri p tion of reality. Indee d, Schrdinger’s equation is a simple linear differential equation, suggesting that it m ight arise as the leading approximation to a more complete model. ” Consistent wit h experience in other areas of physics it is nat ur al to think of QM as emerging in the limit of low energy or long distance. However, it could also emerge in the limit of reduced qua n tum c o mplexity. In other words, QM may not hold for Hilbert spaces of high compl exity. Exp erimental te sts of QM , such as Bell inequality violat ions, have been restric ted to relatively simple and small Hilbert spaces , and problems of low computational compl exity. 153 However, it is not clear how to measu re quantum complexity. It may not be just a que stion of the siz e of the Hilbert space, but perhaps its geometry. For just two enta ngled qubits th e geometry is complex. 155 Independent of experi mental evidence, EmQM provi des an alternative i n terpretat ion to quantum theory that avoids the thorny issues such as the ma ny -worlds in terpretat ion. 8. Quantum cond ensed matter physic s A centra l question of condensed physics is h ow do the physical prop erties of a stat e of matter emerge from the properties of the atoms of which the material is composed and their inte ra cti ons ? There a r e three dimensions to the question. One is at the macroscale to characte rise the physical properties of a specific state of matter. A second dimension is to characte rise the microscopic constituents of ma terials exhibiting this state. The third dime nsion is to connect the macroscopic and microscopic properties. Usually, this connection is done via the mesoscale. Other important questions include whether all possible states of mat ter can b e cla ssified; and if so, how. Eme rgen ce de scribes w hy condensed m atter physics w orks as a unifie d discipline. 32 As a result of uni versality there are concepts and theories that can describe diverse phe no mena in mat erials that are chemically and structurally diverse. For example, although superfluidity occurs in l iquids and superconductivity in solids ther e are uni fy ing conce p ts that can describe both. 8.1 States of matter Condensed ma tter physics is about the study of different states of matter. There are a mult itude of states: liquid, crystal, liquid crystal, glass, s uperconduct or, superfluid, ferromagneti c, antiferromagnetic, quantum Hall, … Each s tate has qualitatively different propertie s. Ev en withi n th is list, there are a m u ltitude of different st ates. For exa mple, there are diffe rent l iquid c rystal states: nematic, chlores tic, s mect ic … This incredible diversity is characte ristic of emergenc e. A state of m atter has emerge n t properties as the constituent parts of the system (atoms or mole cules) do not have these properties. Novel prop ertie s m ay include spontane ous symmet ry breaking, long-r ange order, generalised rigidity, topological defects, and distinct low-energy excitations (quasipa rticles) . Diamond a nd graphite are distinct solid s tate s of c arbon. They have qua lit a tively diffe rent physical properties, at both the microscopic and the macroscopic scale . Graphite is common, blac k, soft, and conducts electricity moderately well. In contrast, diamond i s rar e, transpare n t, hard, and conducts el ectricity very poorly. The ir crystal structures are different and the network of int eractions between carbon atoms are different. In many c ases, symm etry ca n b e used to define the q ualitativ e difference b etween di ff erent state s and th e nat ur e of the orde r ing in the state. An order parameter can be used to quantify this qualitative di ff erence. It only has a non-zero val ue in the ord ered state. S elf -organisation is reflected in the fact that for state s in therm odynamic equilibrium as the system is cool ed below a transition temperature it self -organises into a st ate with the long- range order a ssociated with the broken symmetry. This w hy it is sai d that the symmetry is spontaneously broken. Discontinui ties. A s external paramete rs such as temp erature , pr essure, or magne tic field are varie d discon tinuitie s or s ingularities occur i n thermodynamic properties such as the specific heat capacity or density, as the system undergoes a p hase tra nsition, i.e., transforms into the ordered state. The se discontinuities can be used to m ap out a phase diagram which shows under what c onditions (external parameters) the different states are thermodynamically stable . Universalit y i s evide n t in that materials that have different chemical composition or crystal structure can be found in the same state of matter. Solids are ri gid and it costs energy to d istort the shape of a finite solid with external stresses. This contra sts with liquids whose shape changes to that of it s container. In general, there is a rigidi ty ass ociated with a n ordered state of matter. Disturbing the relative spatial a rr angement of the c onstituents of th e system has an e n ergy cost. This defines an emergent length scale , which is ofte n mesos copic . In superc ondu ctors this scale is the coherence length. Landa u ’s theory of ph ase transitions provides a n e ffective the ory for a ll states of matter associat ed with a broken sym metry. It is a valid at the length scale associate d wi th the generalised ri gidity. Based on the symmetry of the or der paramet er, th e free energy functional a ssociated with the theory can be written down using group theoretical anal ysis. Modularity at the mesoscale. In som e state s of m atter, propertie s ar e det ermined by topologi cal defects such as vortices in superflu ids and dislocations in solids. The spatial size of these de fects is typically of the orde r of the length scale associate d wi th the generalised rigidi ty and muc h larger than the microscopic scale. Metal lic materials that are used in industry and studied by metallurgists are not perfectly ordered crystals. Many of the ir properties, such as plasticity and th e rate of crystal growth from a m elt, ar e dete rmined by the presence and beh aviour of topological defects such as disloca tions and disinclinations. This is an example of how macrosc opic behaviour is largely dete rmined by and best understood in terms of the mesos cale and not the atomic scale. Unpredictabi lity. As discuss ed previ ously, v ery few states of matte r h ave bee n predicted prior to the ir experimental discovery. Thei r existence in principle mi ght be predicted based on Landa u’s theory. However, pred icting whethe r such a state wi ll be exhibited by a mate r ial with a spec ific chemical composition is much ha rd er. Until a few decades ago it was believed that Landau’s theory provided a complete cla ssification of all states of matter. However, it turns out there a r e states of mat ter that do not exhibi t a broken symmetry, but rather topological order. Topological order This is present in sev eral different state s of m atter, some of which wi ll be discussed further below. Exa mples include the quantum Hall effect, H aldane quantum spin chai ns, topological insulat ors, and the Ising Z 2 lattice gauge theory. A signature of topological order in quantum many-body systems i s that the d egeneracy of the ground state de p ends on the topology of space, such as the g enus of the spac e, e.g., for a two-dimensional surface whether it has the topology of sphere , donut, or pr etz el. The genus is a topological invariant, meaning that it is unchange d by con tinuous distortions of the surfac e. The topological characte r of the state mea ns that there are qualitative differences between states and provides a natural explana tion of universality as m any of the details do not matter. An exampl e of an ef fective theory for a state with topological order is the topological field theory of We n and Ze e that includes a Chern-Simons topological term to describe the edge currents in a fr acti on al qua n tum Hall state. 156 Critic al ph enomena The study of systems c lose to a cr itical poi n t in their phase diagram has provided import ant insights int o emergence. Th e renormalisation group (RG) provides a concrete and rigorous tool t o connect scales, construct effective theories, a nd define universality cla sses. The RG is neede d to treat the effect of thermal fluctuations in the local order that are n eglecte d by Landa u’s theory which is a mean -fie ld theory. Fluctuations can lead to qualitativel y d ifferent propertie s. Consider the case of a flu id near t h e liquid-gas critical point . There are a range of different scale s at whi ch the system can be studied: fro m the macroscopic scal e of the whole fluid down to the m icroscopic scale of a toms and molecules. O f partic ular relevance is the mesoscopic scale of blobs of gas insid e liquid. Near the critic al point these blobs increase in size l eading to critical opalescence wher e incident light is scattered sufficiently for the fluid to ta ke on a milky appearance. Wit h the range of scales there is a hi erarchy of effective the or ies: quantum chemistry, va n d er Waals , hard sphere f luid, Ising lattice gas, Landau th eory, and the Wilson (renormali sa tion group) theory. The effective theory at each strata can be derived, or at least j ust ified, from the theory at the strata bel ow. Toy models There i s a long list of toy models that have provided importa n t insights into strongly correlate d electron systems. Examples include th e Anderson model for a magnetic impurity atom in a metal, the Hubbard model for the Mot t insulator-metal transition, the Kondo model for a spin in a metal, the AKLT mod el for a n antiferromagnetic spin-1 chain, th e Luttinger model for a on e-dimensional interacti ng f ermi on g as, and the Haldane model for a topologi cal insulator. Th e emphasis in the de sign of t hese models is simplicity and tractability for analysi s, not faithfulness to the microscopic d etails of real mate r ial s. The k ey attribute of most of these m odels is tha t they exhibit a speci f ic emergent ph enomenon and that this can be turned on and off by va ry ing a pa rameter in the mod el or temperature. Study of several specifi c toy models led to predictions of new states of matter. 8.2 Quasiparticl es In many-body systems the modularity that occ urs at t he mesoscale is exhibited by the existe nce of quasiparticles. They are observable entities and can be the ba s is of effective theori es. Fi gure 3. C artoon re pr esenting the concept of a qu asiparticle in a many-particle system. 157 The c oncept of quasiparti cles is illustrated by an ana logue in Figure 3. When a horse gallops through the desert it stirs up a dust cloud that travels with it. The motion of the horse cannot be separa ted from the accompanying dust cloud. The y act as one entity. Consider a system consisting of m any interacting particles. When one p article moves it carrie s with it a “cloud” of other partic les. This composite entity is referred to as a quasiparticle. O ften it is easie r to understand t he system in terms of the quas iparticle s r ather than the individual particles. This is beca use the quasiparticles interact with one another via an effective interacti on that may be much weak er than the interaction between the original constituent particles. Like the constituent particles in the system quasiparticles have properties such as charge, mass, and spin. However, the se properties for a singl e quasipartic le may be different from those of the individual particles , thus exhibiting the n ovelty characte r istic of emergence. An exam ple is holes in semiconductors; a system of many electrons in a crystal acts collectivel y to produce a ho le (t h e absence of a single electron), a quasiparticle with the opposite charge to tha t of a single electron. More striking examples, such as in the fractional qua n tum Hall effect , are discussed bel ow. Form ulating an effective theory in terms of quasiparticles requ ires profound physical insight. Landa u was one of the first theoretical physicists to take thi s appro ach, i n troducing t h e idea of quasipart icles in his theories of s uperflui dity in 4 He and of li qu id 3 He. Liquid 3 He is c omposed of 3 He atoms that are fermions . In the li quid state they interact strongly with one another. L andau ’s Fermi liquid theory describes the system at temperatures below about 1 K (the degeneracy temperature) in terms of w eakly i nteracting quasiparticles. The quasipa r ticles are fermions , but their effective mass is many times the mass of an isolated 3 He at om. Experiments show that as the pressure increases from 1 bar to the melting pressure of 30 bar, the e ffective mass rat io increases from about 3 to 6. The quasiparticle s are not parti cles: they have a finite lifetime and so th eir e n ergy is not precisely defined, but th e uncert ainty in their energy is much less than their energy . For qua sip articles with the Fermi energy, the un certaint y tends to zero as the temperat ure approac hes zero. The e ffe cti ve Hamiltonian describing Landau’s Fermi liquid only describes low energy exci tations re lative t o the ground state. The energy region of validity is at most the renormalised Ferm i energy, which is related to the d egene racy or “coherence” temperature. The Ferm i liquid in 3 He al so exhibits a new collective excitation zero sound, a density wave predicted by L andau to exi st at frequencies much larger that the quasip arti cle re laxation rate. Landa u’s Fermi liquid theory provid es a justification for the band theory of simple metals, such as elemental meta ls. It jus tifie s th e Sommerfeld and Bloch models, which as a first approxim ation ignore any interaction between the electrons in a metal. The y are s urprisingly successful as the ba re interaction energy du e to Coulomb repulsion i s co mparable to the Fe rmi energy whi ch is a measure of t he kinetic energ y of the elect rons. Fermi liquid theory can be derived by a renormalisation group approach. The quasiparticles and effecti v e inte ractions emerge when high energy degrees of fre edom are “integrate d ou t” i n a functional inte gra l approach. 158 Landa u’s Fermi liquid theory provides an example of adiabatic continuity , which Anderson ident ified as one of the organi sing pr inciples of condensed matter physic s. 27 Consider a Hamil tonian in which the interaction between the fermions is proportional to a v ariable paramete r  . Then as  is increa sed fro m zero to large values there is no qualitative change in the c haracter of the ex citati on spe ctrum. The quantu m numbers of cha rge and spin of the exci tations do not change. In d ifferent words, as  is varied there is no phase transiti on in the ground state of the sys tem. This is not always t h e case. Below, I discuss cases where the quasipart icles (ex citati ons) can h ave different quantum numbers to t h e constituent particles. Universalit y. At tem p erature s be low the degeneracy tempe r ature T F properties of the Fermi liqui d are universa l functions of T/T F. Fe rmi liquid th eory can de scr ibe l iquid 3 He, electron liqui ds in e lemental metal s, and deg enerate Fermi gases of ultracol d atoms. Quasiparti cles are only well-def ined at some e n ergy s cal e, usually a scale much smaller tha n typic al energies in the sys tem . In Ferm i liquids this is known as the coherenc e temperature. This ca n be seen by certain experimental signatures. For example , a Dr ude peak in the optical conduct ivity is only seen for temperatures less than the cohere n ce temperature. 159 Superflui d 4 He . Landau proposed that the low-lying excitations were quasiparticles, which were densit y fluctuations, w ith an e x citation spectrum that was l inea r for sm all wave v ectors (phonons) and had a local minim u m as a func tion of wavevector. He dubbed the excitations associat ed with the minimum as rotons. L ater Feynman int erpreted the roton as a vor tex ring. According t o Wen, 160 there is an effective interacti on between rotons with the same distance depende nce as a dipole-dipole interaction in electrostatics. In a system wit h a broken symmetry associated with continuous symmetry , there are l ow- energy excit ations with a linear dispersion relation. These are Goldstone bosons and their number and c h arac ter reflect the underlying order. j Examples include phonons in crystals and magnons in a ntiferromagnets. Quasi-particles weakl y interact wit h on e anot h er and can be clearl y se en in exp eriments such as inel astic neutron scattering, as shown in the Figure below. 162 On the theory side, the existe nce of quasi-particles is evide n t by the presence of well-defined peaks in spectral densiti es and the lo cation of the maxim u m as a function of frequency and wavevector defines a dispersion re lation, just as would be see n for a non- interacting particle. Exot ic quas iparticles A striking exa mple of the novelty associated with emergence is the existence of quasipart icles th at do not h ave the sa me quantum numbers or stati st ics as the constit u ent parti cles. 163 This is most common in systems of reduced dimensionality (i.e., one or two spatia l dimensions) or with topological order. This show s a breakdown of the adiabatic conti nuity s een i n Landau’s Fermi liquid the ory. Some of these qua sip article s may se em to violate the spin-statistics theorem from quantum fiel d theory. For example, quasipa r ticles with half-integer spin may be bosons. The spin- stati stics theorem conc erns systems of non-interacting particles with L or entz invaria n ce. Then particles with i nteger (half- integer) spin obey Bose-Einstein (Fermi-Dirac) statistics. j The num ber of Nambu- Goldstone mode s (NGMs) c an be lower t h an that of broken symmet ry generators, and the dispersion of NGMs is not nec essarily linear. Examples include the phonons in c rystals of s kyrmions, Wi gner crystals in a ma gnetic field, and spinor BECs. 161 The t heorem does not apply t o condensed matter sys tems as they are comprised of interacting parti cles and th e quasiparticle s ca n have dispersion relations that do no t exhibit L or entz invari ance. Fracti onal Quantum Hall Effect (FQHE) state s. They occur in system s th at are composed of ele ctrons (holes) which have charge - e (+e), obey Fermi-Dirac st atistics, and are confined to move i n only two dimensions in v ery high magnetic fields. However, the charge of the quasipart icles can be a fraction of the charge on a single elect ron , and they do not act as fermions, but have fractional statistics, and are called anyons. An alternative representation of the quasipa r ticle s a re as composite fermions. 164 They are a fermion to whic h is attached an inte ger number of magnetic flux quanta. In th is picture FQHE sta tes w ith fraction 1/ (2n+1) can be viewed as the n- th integer quantum Hall state. Quantum spin chains. These one-dimensional systems of localised spins that couple anti ferro magnetically t o their nearest neighbours are discussed in more detail bel ow. F or spin-1/2, there is no long-range orde r, and the elementary excitations are spinons, which are spin-1/2 excitations tha t obey semion statistics (i.e ., exchange of two particle s produces a phase cha nge of  ). This contrasts wit h three-dimensional antiferromagnet s which exhibit long-range Neel order, and t h e quasiparticles are magnons: bosons with spin-1. Luttinge r liquids . For a system of interacting spinless fermions in one dimension the quasipart icle s in the metallic state are bosons that ca n be ide ntified wi th particle-hole exci tations with a linear dispersion relation. For interacti ng spinfu l fermion systems, in the met allic s tate, spin-charge sep aration occurs. There are two types of bosonic excitations that can m ove independently of one another and with different velocitie s. One carries charge and the ot her spin. It is like the electron (w hich ha s ch arge and spin) has split in two! Majorana fermions. In quantum field t h eory Majorana proposed in 1937 a fermion th at c ould be it s own anti-p article, but such an eleme n tary particle ha s nev er been observe d. Toy mode ls have be en key to proposals that M ajorana’s vision might be reali sed in conde nsed matter. Kitae v studied a one-d imensional lattice model for a topological superconductor and showed that the boundary states were Majorana fermions. 165 It was then suggeste d this might be reali sed in hybrid semiconductor -superconduct or qu antum wire structures. Several experi mental groups subsequently claimed to ha v e obs erved signa tures of Majorana’s. However, these c laims are controversial sinc e there are alternative more mundane expla nations for the experimental observa tions and the cases reported may suffer from selec tion bias. 166 Kitaev also studied a toy spin model on the honeycomb lattice with highly anisotropi c Heisenberg interactions and show ed that it was exactly soluble with a spin liquid ground state and Ma jorana fe r mion quasiparticle s. 167 In spite of e xtensive theoretical and experi mental w ork it is a n ou tstanding que s tion as to whether Kitaev’s model is relevant any real mat erial. 168 A quasiparticle conjecture. Over time it has be en found that for almost every model Hamil tonian with physically realistic interactions there is a way of rewriting the system (performing a un ita ry transformation) so it can be viewed as a system of weakly interacting quasipart icles, ev en when the bare interactions are strong. This raises the question as to whether it is possible to prove som e general (existence) theorem along s uch li nes. 8.3 Superconductivity Superconduct ivity is a poster child for eme rg ence. Novel ty. D istinct properties of the superc ondu cting state include zero resisti v ity, t h e Meissner effect , and the Josephson effect. The normal metallic state does not exhibit the se prop ertie s. At low tem peratures, solid tin exhibits the property of superconduct ivity. However, a single atom of tin is not a superconductor. A sma ll numbe r of tin at o ms has an ene rgy g ap due to pairi ng interactions, but no t bulk superconductivity. Th ere is more than one superconducting state of matter. The order parameter may have the same symm etry as a non-tr ivial representati on of the crystal symmetry, and it can have spin singlet or tr iplet symmetry. Type II superc ondu ctors in a magnetic field have a Abrikosov vortex l attice, ano ther distinct state of matter. Unpredictabi lity. Even though the underlying la ws d escribing the interactions bet ween ele ctrons in a crystal have been known for one hundr ed years, the discovery of superconduct ivity in many specific materials was not predicted. Even after the BCS theory was worked out in 1957, the di scov ery of superconductivity in intermetallic compounds, cuprate s, organ ic c harge transfer salts, fullerenes, and heavy fermion compounds was not predicted. Eve n wi th adva nces in electronic structure theory for weakly correlated electon systems prediction of new superconduc ting materials is rare. 169 Order and structure. In the superconducting stat e the ele ctrons become ordered in a particular way. The m otion of the electrons relative to one another is not independent but correlated. Long range orde r is refle cted in the generalised rigidity which is responsible for the zero resistivi ty. Proper tie s of ind ividual atoms (e.g., NMR chemical shif ts) are diffe r ent in vacuum , in the metallic state, and superconducting state. Universalit y. Proper ties of superc onductivity such as zero electrica l resistance, the expulsion of magne tic fields, quantisation of magnetic flux, and the Josephson effects are universal. The existe nce and description of these properties is independent of t h e chemical and struc tural deta ils of the material in which the superconductivity is observed. This is why Gin zburg- Landa u th eory works so well. In BCS theory, the temperature dependences of thermodynamic and transport proper tie s are g iven by universal fu nctions of T/T c where T c is the transition tem perature. Experimental data i s consistent with thi s for a wide range of superc onducting mat erials, par ticularl y elemental metals for which th e electron-phonon coupling is weak. Modularity at the mesos cale. Emerge n t entities include Cooper pai rs and vor tice s. Th ere are two associat ed emerge n t length scales, typicall y much larger than the microsc opic scale s define d by the interatomic spacing or the Fermi wavelength of electrons. The first scale is the coherence le ngth ; it is associated with t h e energy cost of spatial variations in the order paramete r. It defines the extent of the proximity effect where the surface of a non - superconduct ing metal can become superconducting when it i s in electrical contact with a superconduct or. The coherence length turns out to be of the order of the size of the Cooper pairs in BCS the ory. Th e second length scale i s the magnetic pen etration dept h ( also known as the London length ) t h at determi n es the extent that an exte rn al magnet ic field can penetrate the surfac e of a superconductor. I t is determi n ed by the superfluid density. The relative size of the c oher ence lengt h and the penetration depth determines whethe r an Abrikosov vortex lat tice is stable in a large ma gn etic field. Quasiparticles. The elementary excita tions are Bogo liubov quasiparticle s that are quali tatively different to particle and hole excitations in a normal metal. They are a coherent superposition of a particle and hole exc itation (relative to the Fermi sea), have zero charge and only exi st above t h e energy ga p. The mixed particle-hole character of the quasiparticles is refle cted in the phenomenon of A ndreev reflection. Singularities. Superconductivit y is a non -perturbative phenome non. In BCS th eory the transit ion temperature, T c , and the excitation energy gap are a non-a n alytic function of the ele ctron-phonon coupl ing constant  .     󰇛   󰇜 A singular struct ure is also evident in the propert ies of the current-current correlation function. Intercha nge of the limits of zero wave v ector and zero frequency do not commute, this bei ng in timately connecte d wi th the non-zero superfluid de nsi ty. 170 Eff ective theor ies. These a r e illustrated in t h e Figure below. The many-particle Schrodinger equat ion describes electrons and atomic nuclei interacting with one another. Many -body theory ca n b e used to j ust ify consideri ng the electrons as a jellium liquid of non-in teracting fermions i nteracting with phonons. Bardeen, Pines, and Frohlich showed that for that system there i s an effective i nteraction between fermions tha t is attractive. The BCS theory includes a truncate d version of this attractive interaction. Gorkov show ed that Ginzburg-Landau theory coul d b e deri v ed from BCS theory. T h e London equati ons c an be derived from Gi nzburg- Landau theory. The Josephson equations o nly include the ph ase of order parameter to descri be a pair of coupled superconductors. The fu ll quant u m version of the Josephson equat ions, such as those that describe th e Cooper pair boxes and transmons 171 used in quantum computing, can be derived from BCS theory using a functi on al i n tegral approac h. 172 Fi gure 4 : A hierarchy of effective theories for superconductivity. The historical d evel op ment of t h eories mostly went in the downwards direction in the Figure. London prece d ed Ginzburg-Landau which precede d BCS theory. Today for specific material s where superconduc tivity is know n to be due t o e lectron -phonon coupling a nd the elec tron liqui d is weakly correlated one can now work upw ards using computa tional methods such as Density F unctional Theory (D FT) for Superconductors or the El iashberg theory with input paramete rs calculated from DFT-based met hods. H o wever, this has had debata ble s uccess. 173 The superc onducting transition temperatures calculated typicall y v ary with the approxim ations used in the DFT, such as the choice of functional and ba sis set, and often differ from e xp erim ental results by the order of 50 p ercent. This illustrates how hard prediction i s for emergent phenomena. Pote ntial and pitfalls of mean-fi eld theory. Mean-field approximations and theories ca n provide a useful guide as what emergent properties a re possible a nd as a starting point to map out properties such as phase di agrams. For some systems and properties, they work inc r edibly well a nd for oth ers they fail spectacularly and are mis lea ding. Ginz burg- Landau theory and BCS theory are both mean-f ield theorie s. For three-d imensi onal s uperconductors they work extremel y well. H owever, in two dimensions as long-range order and breaking of a conti nuous symmetry canno t occur a nd the physics associated wit h the Be r ezinskii- Kosterlitz - Thouless transition occurs. Nevertheless, the Gi n zburg-Landau theory provides the background t o understand the justificati on for the XY model and the presence of vor tices to proceed. Si milarly, the BCS theory fails for strongly correlated electron systems, but a version of the BCS theory does give a surprisingly good description of the superconducting state. Cross -ferti lisation of fields. Concepts and methods developed for the theory of superconduct ivity bore fruit in o ther sub-fields of physics includi ng nu clear physics, ele mentary particles, and astrophysics. Considering the matter fields (associa ted with the ele ctrons) coupled to electromagne tic fields (a U(1) gauge the ory) the matte r f ields can be inte gra ted out to give a theory in which the photon has mass. This is a di fferent perspective on the Mei ssner effect in which the magnitude of an external magnetic f ield decays exponent ially as it penetrates a superconductor. This idea of a massless gauge field ac quiring a ma ss due to spontaneous s ymme try breaking was central to steps towards the Standard Model ma de by Nambu 174 and by Weinberg. 8.4 Kondo effect Th e system consists of a single magnetic impurity in a metal wit h an antiferroma gn etic coupli ng between the spin of the impurity and all the spins of the electrons in t h e metal. In real mat erials the system is a dilute solution of m agnetic atoms in a non-magnetic metal. Novel ty. The ground sta te is qual itatively differe n t from the high temperature (disordered) state or if the m agnetic interaction is not present. The ground state i s a spin sing let and has no net magne tic moment. In contrast, the disordered st ate of a Kondo system is degenerate a nd h as a net m agnetic moment, seen by a Curie magnetic susc eptibilit y at high tem p erature s. Emergent ene rgy sca le. The Kondo tem p erature T K is much smaller than the “bare” energies in the system: the anti ferro magnetic exchange coupling J and the Fermi energy of the metal. Exp erimentally, the sca le of T K is evident from the temperature at which ther e is a minim u m in the resistivity and the magnetic susceptibility deviates from Cur ie-like beha v iour. In multi-channel problem s there can be a hierarchy of en ergy scales, corresponding to separate screening of orbita l and spin degrees of freedom. Singularit ies . The Kondo effe ct is n on-perturbative. A perturbation theory in powers of the interacti on J gives expre ssions for physical quantitie s such as the resistivity that diverge as the temperature goes to ze ro. This is inconsistent with experiment w hich show physical qua ntities are finite and vary smoot hly as the temperature decreases. Th is was known as the “Kondo problem ”. The non-pert urb ative beha viour is also seen in that t he calculated Kondo temperature as a function of J has an e ssential singularity as a functio n of J. The system e xhibits asymptotic freedom (borrowing termi no logy from QCD) in tha t as the energy scale decrea ses the effective coupling increases, becoming infinite at zero energy. Discontinui ty There i s no phase transition, i.e., all physical quantiti es are smooth functions of tempera ture. This shows that novelt y do es not necessarily imply discontinuity. Eff ective theor ies The Kondo mode l Hamiltonian can be derived from the Anderson impurity m odel that allows charge fl u ctuations on the impurit y atom. The Schrieffer-Wolff transformation can be used to inte gra te out charge de gr ees of freedom in the Anderson model and obt ain the Kondo model as an effe ctive theory. In the conti nuu m limit the Kondo model can be mapped to boundary conformal theory. T h is approach ha s be en particularly fruitful for i nv estigating multi- channel Kondo problems (where t here are orbi tal degrees of freedom) some of which has a non-Fermi liqui d ground state. 175 In some of these models there are hie r archie s of multiple energy scales. Universalit y The t emperature dependence of the contribution of the magnetic impurity to thermodynamic an d transport properties are universal functi ons of T /T K. Experimental data for a wid e range of impurit y atoms and metals w ith diverse value s for J and band structure parameters collapse onto single theoretical curves. The Sommerfeld-Wi lson ratio of t h e magnetic susceptibility to specifi c heat coefficient has the universal value 2, compared t o the value of 1 for S ommerfeld and Bloc h models for w eakly correlated Fe rmi liquid s . The Kondo effect has also been observed in artificial systems including semiconductor quantum dot s, and magnetic atoms and mol ecules on m etallic surfaces. The experimental data collapses onto the same universal scali ng functions. 8.5 Quantum sp in chains Novel ty Wit h antiferromagnetic nearest-ne ighbour interactions the spin-1 c hain has properties quali tatively different from a three-dimensional syste m. The ground state has no long-range order and the r e is a fi nite energy gap to the lowest energy spin triplet excitation. There i s topologi cal order and there are edg e excitations with spin-1/2. The latter is an example of fracti on ali sa tion of the quantum numbers of the quasi-particles as the system is c o mposed of constit uents with spin 1. Chains of non-inte g er spins can be described by an e ffective the ory which is the non -l inear sigma m odel including a topological term in the action. This term changes the physics signific antly from integer spin sys tem s. There is no symmetry breaking and qua si -long-ra ng e order: spin c orre lations exhibit power la w decay. There are gapless spin -1/2 excitations in the bulk and t hese are semions. Fi gure 5. Hi erarchy of objec ts and effective theories associated with a chain of atoms with a loca lised spin-1. Toy model The AKLT model is exactl y solub le and capture s key phenomena se en in spin-1 chains: the energy gap, t opo logical orde r, short-rang e spin correlati ons, and spin-1 /2 edge excitations. 8.6 Topological insulators Novel ty A topologic al insulator is a distinct state of e lectronic matter with no spontaneous symmetry breaking. It is a time reve rsal invariant and has an el ect ronic band gap in the bul k. Transport of charge a nd spin o ccurs by gapless edge st ates. Thi s state of m atter is associate d wi th a novel t opological invariant, which distinguishes it from an ordinary insulator. This invariant is anal ogous to the Chern number used to classify states in the integer quantum Hall effect. 176 Toy models Haldane hi gh lighted in his Nobel Lect ur e in 2016, that toy models played a central role in the discovery of t opological insulators in real materials. In 1988, Haldane published “ Model for a Quantum Hal l Effect without Landau Levels: Condens ed-M atte r Realization of the "Parity Anomaly" ” . 177 The model was a tight-binding model for spinless fermi ons on a honey comb net with alte rnating magne tic fl ux through plaquettes. He noted a phase transition as the magne tic flux was varied. Haldane concluded his paper with the comment, “ the particular model presented here is unlikely to be d irectly physicall y r ealizable. ” Haldane’s paper recei v ed little attent ion until 2005. Following the fabrication of graphene , K ane and Mele, published a paper, “ Spin quan tum Hall effect in gra p hene ,” 102 suggesting how the state proposed by Haldane could occur. How ever, t h eir estimate of the spin-orbit coupling coupli ng ne eded was several ord ers of magnitude larger than reality. N everthele ss, the two- dime nsional topolog ical insulator state was subsequently proposed by Bernevig, Hughes, and Zhang 179 to exist in quantum wells of mercury tell ur ide sandwiched between ca d mium tel luride, and then observed. 180 8.7 Quantised properties There a r e two characte r istic s associated with states of matter being referred to as “ quantum. ” One chara cteristic when properties are determined by the quantu m statistic s of par ticles, i.e., the system i s composed of part icles that obe y Fermi-D irac or Bose-E instein statistics. The second cha racteristic is that a macrosc op ic property is quantiz ed with values determined by Planck’s consta nt. I now discuss ea ch of the se with respect to emergence. A. Quantum statisti cs For a syste m of non- interacting k fermions and bosons at hi gh temperatures the properties of the system are those of a classical ideal gas. As the temperature de creases there is a smooth crossover to low-tem p erature prop erties that are qualitatively different for fermions, bosons, and cl assical particles. This crossover occurs around a tem p erature, known as the degeneracy tem perature, that is dependent on the particle de nsi ty and Planck’s consta n t . Many of the properties resulting from qua ntum statistics also occur in system s of strongly inte ra cti ng p article s and this is cent r al to the concept of Landau’s Ferm i liquid and viewing liqui d 4 He as a boson liquid. If liquid 3 He and the ele ctron l iquid in e lemental metals are viewed a s a gas of non-in teracting fe r mions, the deg enerac y temperature is about 1 K and 1000 K, respecti vely. Ther modynamic properties are qualitati v ely di ff erent above and below the de generacy temperature. Low-temperature properties can have values that differ by orders of magni tude from c lassica l values and h ave a different temperature depe nd ence. In contrast to a c lass ical ideal gas, a fermion gas has a non-zero pressure at ze ro tempe r ature and its magni tude is determined by Planck’s constant . Th is degeneracy pressure is responsible for the gravita tional stability of white dwarf and neutron stars. Th ese properties of systems of particles can be viewed as emergent properties, in the sense of novelt y, as th ey are qu alitatively different from high-temperature properties. However, they involve a cross over as a funct ion of temperature and so are not a ssociated with discontinuity . They a lso ar e not associ ated with unpr edictability as they are straight-forward to calculate from a knowle dge of microscopic properties. B. Quantised macroscopic properties These provi de a more dramatic illustration of e mergence. Here I c onsider four spe cific systems : superconducting cylinders, rotating superfluids, J osephson junctions, and the inte g er Quantum Hal l effect. All th ese systems have a macros copic property that is observed to ha ve the fol lowing features . As an external p arameter i s varied the quantity def ining a macroscopic property varies in a step-like manne r wi th discrete values on the steps. This contrasts to the smooth linear varia tion seen when the material is not cond ensed in to the quantum state of matter. The va lue on the steps is an integer mul tiple of some specific p arameter. This para meter (un it of quantisati on) on ly depends on Planck’s constant h and other fundamental constants. The uni t of quantisation does not depend on det ails of the material, such a s chemical compositi on, or d etails of the de vi ce, such as its geometrical dimensions. The quantisati on h as been observed i n d iverse materials and devices. Expla nation of the quantisation involves topology. k Simil ar behaviour occurs for weakly interacting particles. N on-interacting particles obeying quantum statistics can be viewed as interacting in the sense that the statistics provides a constrai nt concerning two particle s be ing in t h e same quantum state (e n ergy le vel) and so could be viewed as an attractive interaction for bosons and an infinitel y r epulsive interacti on for fermions. Superconduct ing cylinders . A hollow cylinde r of a metal is placed in a magnetic field parallel to the axis of the cylinder. In the metallic state t he magnetic flux e n closed by the cylinder increa ses linearly with the magnitude of the external magnetic field. In the superconducting state , the flux is quan tized in units of the ma gn etic flux quantum, Φ 0 = h /2 e wh ere e is the charge on a n ele ctron. It is a lso found that in a type II superconductor the vortices that occur in the presence of an e x ternal magnetic field enclose a magnetic flux equa l to Φ 0. Rotat ing superfluids. When a cylinder contai ning a normal fluid is rotated about an axis passing down the ce ntre of the cylinder the fluid rotates with a circulati on propor tional to the speed of rota tion and the diameter of the cylinder. In contrast, in a superflui d, as the speed of rotat ion is varied the circulation i s quan tised in units of h / M where M is the mass of one at om in the fl u id. This quantity is also the circulation around a single vortex in the superfluid. Josephson junct ions. In th e metallic state t h e current passing through a junction i n crease s line arly w ith the voltage a pp lied across the j unction. In the superconducting state t he AC Josephson effect occ urs. If a be am of microwave s of constant frequency i s in cident on the junct ion, jumps occur in the current when the voltag e is an integer multi p le of h/2 e. The quanti sation is observed to b etter than one p art i n a million (ppm). Integer Quantum Hall e ffect . In a normal conductor the Hall resistance increa ses l inea r ly with the e xternal magnetic field for small magnetic fields. In contrast, in a two-dimensional conduct or a t high magnetic fi elds the Hal l resistance is quantized in units of h /2 e 2 . The quanti sation is observed to better than one part in ten million. Reflecting universalit y, the observed val ue of the Hall resistance for each of the plateaus is indepe nd ent of ma ny d etails, incl uding the temperature, the amount of disorder in the material, th e chemic al composition of system (sili con versus gall ium arsenide), or wheth er the c h arge c arriers are electrons or holes. Macroscopic quantum effect s are also seen in SQUID s (Superconducting Quant um Interfere nce Devices). They exhibit quantum interference phenomena a na logous to t h e double-slit experiment . The electrical current passing through the SQU ID has a periodicity define d by the rat io of the magnetic flux inside the c urrent loop of t h e SQUID and the quantum of magnetic flux. The precision of t he quantisation provides a means to accurately determine fundamental constant s. Indeed, the title of the pa p er announcing the discovery of the integer quant u m Hall effect was , “ New Method for High-Accur acy Determination of the Fine-Structure Constant Based on Quanti zed Hall Resistance .” 181 It is astonishing that a macroscopic measurement of a property of a macroscopic system, such as t h e electrical r esistance, can determine fundamental constants that are normally associated w ith the microscale and properties of atom ic sys tems. Laughl in and Pines claim ed that the quantisation phenomena described above reflect organizi ng pr inci p les associa ted with emergent phen omena, a nd thei r un iversality s upports thei r c laim of the unpred ictability of emergent properties. 61 , l Quantum states of matte r and metrology The unive rsa lit y of the se m acroscopic quantum effects has practical applicat ions in met rology, the study of measurement and the associated unit s and st andards . In 1990 new inte rna tional standards were de f ined for the units of voltage and electrical resistance, based on the qua ntum Hall effect and the AC Josephson effect, respectively. Prior to 1990 the standard used to define one volt was based on a part icular type of electrical batt ery, known as a Weston cell. Th e new standa rd us ing the AC Jos ephson effect allowed volta ges to be defined with a precision of better than one part per billion. This change was moti vated not only by improved precision, but also improved portability, reproducibility, and flexi bility. The old voltage standard involved a specific material and de v ice and required maki ng duplicate copies of the standard Weston cel l. In contrast, the Josephson voltage standard i s independent of the specific materials used and the details of the de v ice. Prior to 1990 the i n ternational standa rd for the ohm w as define d by the electrical resistance of a col umn of liquid mercury with constant cross -sectional area, 106.3 cm long, a mass of 14.4521 grams and a temperature 0 °C. Like the Josephs on volta ge standard, the quantum Hall resist ance standard has the advantage of precision, portability, reliability, reproducibilit y, and inde p endence of p latform. Th e independence of the new voltage and resistanc e standards from the platform used refl ects the fact that the Josephson and quantum Hall effects have the universality characteristic of e merge nt phenomena. 9. Classical condensed matter l They sta ted “ Th ese thi ngs are clearly true, yet they cannot be deduced by direct calc u lation from the T h eory of Everyt h ing, for exact re sul ts cannot be predicted by approximate cal culations. This point is s til l not understood by ma ny professional physici sts, who find i t easie r to believe that a deductive link exists and has only to be discovered than to face the truth t hat there is no link. But it is true nonetheless. Experiments of this kind work because there a r e highe r organizing principles in nature that make them work. The Josephson quantum is exact because of the principle of continuous symmetry brea k ing (16). The quantum Hall effect is exact because of lo calization (17). Neither of t h ese things ca n b e deduce d from microscopics, and bot h are transcende nt, in that the y would continue to be true and to l ead to exact results even if the Theory of Eve rything were c h anged. Thus t he existe nce of these effects is profoundly important, for it shows us that for at lea st some fundamental things in nature the Theory of Everything is irrelevant. P. W. Anderson’s famous and a pt description of this state of affairs is ‘‘more is different’’ (2). The emergent physical phenomena regulated by higher org anizing principles have a property, namely t h eir insensiti vity to m icroscopics, that is directly relevant to the broad question of what is knowable i n the deepest sense of the term. ” 9.1 Continuous p hase transiti ons in two dimensions In two dime nsions the ph ase transition that occ urs fo r s uperfluids, superc onductors, and plana r c lassical magnets is qualitativel y d ifferent fro m tha t whi ch occurs in higher dime nsions. K nown as the Berezinskii-Kosterlitz-Thouless transition, it involves several unique emergent ph enome na. Novel ty The l ow-te mperature state doe s not exhibit long -range-order or spontaneous symmetry breaking. Instead, the order parameter has power law correlations, bel ow a temperature T BKT. Hence, i t is qualitatively different from the high-te mperature disordered state in which the correlati ons decay exponentially. It is a d istinct state of matter, m with properti es that are inte rmediate be tween the low- and high-temperature states normally a ssoci ate d wi th phase transit ions. Th e power la w correlations in the low-temperature state are similar to those at a convent ional critical point, which decay in powers of the c ritical exponent  . However, here the pha se diagram can be viewed as having a line of critical points , consisting of all the tem peratures below T BKT. Along this line the critical exponent  var ies cont inuously with a value that depends on interaction strength. In contras t, at c onve ntional critical points  has a fixed va lue det ermine d by the uni v ersality class. For example, for the Ising model in two dime nsions  =1/4, ind epende n t of the interacti on strength. The m echanism of the BKT phase transition is qualit atively diffe r ent from t h at for convent ional phase transitions. It is driven by the unbinding of vortex and anti -vortex pairs by therm al fluctuations. In contrast, conventional phase transitions are driven by thermal fluct uations in th e ma gn itude of the order pa r ameter. Discontinui ty There i s a discontinuity in the stiffness of the order parameter at the transition temperature, T BKT. The specific h eat capacity i s a continuous function of temperature , in contrast to the singulari ty that occurs for conventional phase transiti ons. Toy model A classica l Heisenberg model for a planar spin, a lso known as the XY model, on a two- dime nsional square lattice captures the essential phy sics. Modularity at the mesoscale The quasipa r ticles of the system that are relevant to understanding the BKT phase transition and the low-t emperature st ate ar e not magnons (for magnet s) or phonons (for s uperfluids), but vorti ces , i.e., topological de fects . These entities usually occur on the mesoscale. The releva n t effe ctive theory is not a non -linear sigma model. Thermal excitati on of vor tex- anti vortex pairs determine the temperature dependence of physical properties and the transit ion at T BKT. There is an effective inte rac tion between a vortex and anti-vortex that is att ra cti v e and a logarit h mic function of their spatial separa tion, analogous to a two- dime nsional Coulomb gas. Universalit y m Sometimes it is stated that the low -temperature state has topological order, but I am not reall y sure what that means. The BKT t ransition oc curs in diverse ra ng e of two-dimensional models and materials incl uding superf luids, superconductors, ferroma gnets , arrays of Josephson junctions, and the Coulomb ga s. The ratio of the discontinuity in the or der paramete r st iffness at T BKT , to T BKT has a universal v alue, independe n t of the coupling st r ength. The renormal isation group ( RG ) e quations associate d with the t r ansition are the same a s those for a multit ude of other systems , both cla ssic al and quantum. The classical two- dime nsional s ystems include t h e Coulomb gas, Villain model, Z n model for large n, solid-on- solid model , eight v ertex model, and the Ashkin-Teller model. They also apply to the cla ssical Ising cha in with 1/ r 2 inte ractions. Quantum models with the same RG equations incl ude the an isotropic Kondo mod el, spin boson model, XXZ antiferromagnetic Heisenberg spin chai n, and the sine-Gordon quantum field theory in 1+1 dime nsions. In oth er words, all these m odels are in the same universality class. Singularit y The correlation le ngth of the orde r p arameter is a non -analytic function of the tem p erature.  󰇛  󰇜   󰇛       󰇜 This essenti al singularity is re lated to the non-p erturbative nature of the corresponding quantum models. Two-dimensional crystal s Rela ted phys ics is relevant to the solidification of two-dimensional liquids, and the associated theory was deve loped by Halperin, Nelson, and Young . The relevant toy model is not the cla ssical X Y model as it is nece ssary to include the e ffect of t he discrete rotational symmetry of the lattice of t he solid. The low-tem p erature state exhibit s discr ete ro tational, but not spatia l, sym met ry br eaking, with power law spatial correla tions. This state does not directly mel t into a liquid, but int o a distinct state of matter, the hexatic pha se. I t has short-range spati al order and quasi-long-range orientational (sixf old) order. The phase transitions are driven by t opological defects, disclinations and dislocations. Predic tability The BKT t ransition, th e quasi-ordered low tem p erature state , and the hexat ic phase were all predicted t heoretically before they were observed experimentally. This is unusual for eme rgen t phenomena but shows that unpred ictability is not equivalent to novelty. 9.2 Spin ices Spin ice s are magnetic materials in which geometrically frustrated magnetic interactions betwee n the spins prevent long -range magnetic orde r and lea d to a residual ent ropy similar to that in ice (sol id water). Spin ices provide a bea u tiful example of ma ny aspect s of emergence, incl uding how surprising new entities can emerge at the mesoscale. Novel ty Spin ice s are composed of individual spins on a lattice. The system exhibits properties that the i ndividual spins and the high- temperature st ate do not have. Spin ice s exhibit a novel state of mat ter, dubbed th e magnetic Cou lomb pha se. There is no long-range spin order, but there are power-la w (dipolar) correlations that fall off as t he inverse c ub e of dista nce. N ovel enti ties include spin defects reminiscent of magnetic monopoles and Dirac strings. The novel propertie s can be understood in terms of an emerg ent gauge field. The gauge theory predicts that the spin correlation function (in momentum spac e) has a pa rticular singular form exhibi ting pinch points [also known as bow ties], wh ich are se en in spin-pol arised neut ron scatt ering exper iment s. 182 , 183 Toy models Classica l models such as the Ising or Heisenberg models with antiferromagnetic nearest - neighbour i nteractions on the pyrochlore lattice exhibit the emergent physics associated wit h spin ice s: absence of long-rang e order, resi du al entropy, ice type rules for lo cal orde r, and long-range di po lar spin c orrelations exhibiting pinch points. These t oy models can be used t o derive the gauge theories that describe emergent properties such as monopoles and Dirac strings. Actual materials that exhibit spin ice physics such as dyspros ium t itanate (Dy 2 Ti 2 O 7 ) and holmi um titanate (Ho 2 Ti 2 O 7 ) are more complicate d t han these t oy models. Th ey involve quantum spins, ferromagnetic interactions, spin-orbit coupling, c rystal fields, complex crystal structure a nd d ipola r magnetic i n teracti ons. Henl ey states 184 that these materials "are well approxim ated as having nothing but (long-ranged) d ipolar spin interactions, rathe r than nearest-neighbour ones. Although t h is model is cle arly related to the “Coulomb phase,” I feel it i s largely an independent paradigm with its own concepts that are different from the (entropic ) Cou lomb pha se..." In different words, the classical Heisenberg models are toy model s that illustrate what is poss ible but should not be viewed as effecti v e theories for the act ual materials exhibiting spin ice physic s. Eff ective theory Ga uge fields described by equations analogous to electrostatics and magnetostatic s in Maxwell’s theory electroma gnetism are emergent in coarse-grained descriptions of spin ices. I briefly de scribe how this arises. Consider an antiferromagnet ic Heisenberg model on a bipartite lattice where on each latt ic e site the r e is a tetrahe dron. The "ice rule s" require that two spins on each tetra h edron point in and two out. Define a field L (i) on each lattice site i which is the sum of all the spins on the tet rah edron. The magnetic field B ( r ) is a coarse graining of the field L (i ). The ice rules and loca l conservation of flux require that      The ground sta te of this model is infinitely degenerate w ith a re sidu al entropy S 0. The eme rgen t “magnet ic” field [which it should be stressed is not a physical magnetic field] all ows the presence of monopoles [magnetic charges]. These correspond to de f ects that do not sati sfy the local ice rules in the spin system. It is argued that the total free e nergy of the system is K is the "stiffness" or "magneti c permeability" associated wit h the ga ug e fiel d. I t is entirel y of entropic or igin, j ust like the elasti city of rubber. n n Aside: I am c urious to see a calculat ion of K from a mic roscopic model and a n e stimate from A local spin flip produces a pair of oppositely charged monopoles. T h e monopoles are deconfi ned in that they can move freely through the l att ice. They are joined together by a Dirac stri ng. Th is contrast s with r eal magnetism where there are no ma gn etic charges, only magne tic dipoles; one can view magnetic charges as confined within dipoles. There a r e only short-range (n earest neighbour) direct interacti ons be tween t he spins. However, these a ct together to produce a long-r ange interaction between t h e monopoles (which are deviations from local spin order). Th is effective interaction betwee n the two monopole s [charges] has the same form as Coulomb’s law.  󰇛   󰇍 󰇍 󰇍     󰇍 󰇍 󰇍  󰇜       󰇍 󰇍 󰇍     󰇍 󰇍 󰇍   This is why this stat e of matter is c alled the Coulomb phase. Universalit y The nove l properties of s pin ice occur for both quant um and classical systems, Is ing and Heisenberg spins, and for a range of frustr ated lattices . S imilar physics occurs with water ice, magne tism, and charge order. Modularity at the mesoscale The system can be understood as a set of weakly interacti ng modular units at different scale s . These i nclude the tetrahedra of spins, the magnetic monopoles, and the Dirac strings. However, it should be noted that Henley states that D irac stri ngs are "a nebulous and not very helpful notion when applied to the Coulomb phase proper (w ith i ts smallish polarisation), for the stri ng 's path i s not we ll de f ined... It is only in an ordered phase... t h at t he Dirac string has a cl ear meaning." 184 The m easured temperature dependence of the specific heat of Dy 2 Ti 2 O 7 is consistent wi th that cal culated from Debye-Hücke l the ory for d econfined charges interacting by Coulomb's law . 182 Unpredictabi lity Most new states of ma tter are not predicted theoretically. T h ey are discovered by experi mentalists, often by serendipity. Spin ice and the magnetic Coulomb phase is an exce ption. Sexy magnetic monopoles or boring old electrical charges? In the di scussion above the "magnetic field" B(r) m ight be replaced with an "electric field" E(r). Then t h e spin defe cts are just analogous to electrical charges and the "Dirac strings" becom e like a polymer chain with opposite electrical charges at its two ends. Thi s is not as sexy. On the other ha nd, it can be argue d that the em ergent ga ug e field is " magne tic". It describe s spin defects and these are associated with a local magnetic mome n t. Furthermore , experi ment. Henley points out that in w ater ic e the e ntropic e lasticit y ma kes a contribution to the di electric const ant and this "has been l ong known." The references given 185,186 calc ula te the physical dielectric constant and the entropic cont r ibution to K is not clear. the l ong-rang e dipolar correlati ons (with ass ociated pinch point s) of th e gauge field a r e dete cted by magnetic neutron scattering. Emergent gauge fields in quantum m any-body systems ? The a nalogy of s pin ices with cla ssical magnetostatic s raises the hope t h at there ma y b e condensed m atter systems that exh ibit eme rg ent classical electromagnetism includi ng eme rgen t photons. 187 Recently, Smith et al., 188 argue d that qua n tum spin ices, including Ce 2 Zr 2 O 7, may ha v e eme rg ent “photons” and spinons. The e xistence of emergent gauge fields in quantum states of matter has been investigated exte nsively by Xiao- Gang We n. He h as shown how c ertain mean-field treatments of frustrated a n tife rromagnetic quan tum Heisenberg models (with quantum spin liquid ground state s) and doped Mott insulators (such as high T c cuprate superconductors) lead to emergent gauge fi elds. A s fascinati ng as this is, there is no definitive evidence for the se emergent gauge fi elds. They just provide appealing theoretical descriptions. This contrasts with the eme rgen t gauge f ields for spin ice, where the pinch points in the neutron scatte r ing spectrum are vie wed as a “smoking gun.” B ased on his success at constructing the se emerge n t gauge fiel ds Wen has p romoted the provocative idea that the gauge fields and fermions that are considere d funda ment al in the standard model of par ticle physics ma y b e emergent enti ties. 160,189 I ment ion two more definitive examples of an emergent gauge f iel d in condensed matter. One is that associated with the Berry curvature that cause s the anomalous Hall effect in metallic ferromagnets. The ga uge po tential is defined in term s of elec tronic Blo ch state s and the Berry curvature is only non-zero i n when time-reversal symmetry is broken. 190 The sec ond example concerns sys tem s with qu asiparticles that are We y l fermions, i.e., gaple ss particles with a definite chirality. In th e A phase of superfluid 3 He , there are topologi cally stable nodes on the Fe rmi surface at the points defined by the vector par allel to the di re cti on of the orbital a ngu lar m o mentum of the order paramete r. Analogous to ele ctromagnetism the vector acts as a gauge potential and its s patial or time-dependent varia tions of the order pa r ameter (such as occur in vortices) define “magnetic” and “electric” fiel ds leading to a chiral anomaly. 191 9.3 Soft matter Soft mate r ial s and st ates of matte r that are “squishy” include polymers, membranes, foams, coll oids, ge ls, and liquid crystals. Soft matter i s ubiquitous in biological materials . This field has provid ed new insights into the physics of biological syste ms and of the science and tec hnology of food. Seminal contributions made by Pierre de Genne s were recognized with a Nobel Prize in 1991. He began his career working on superconductivity and went on to develop a un ified framework t o understand soft matt er, int roduc ing ideas from conde nsed mat ter physics such as effective theories, order paramete rs, sca ling, renormali sa tion, a nd universal ity. McLei sh identified s ix characte r istic s of soft m atter. 192 I list t hem as they relate to characte ristics of emergence. 1. Therm al motion Soft mat ter sys tems exhibit large local spatia l rearrangements of their microscopic constit uents under thermal agitation. In contrast, "hard" mate rials experience only small distorti ons due to thermal motion. Th ese large rearrangement s lead to emergent length scales. 2. Structure on i n termediate l ength scales There a r e basic units (" fundamental" struct ures), typi call y involving a very large numbe r (hundreds to thousands) of at oms, that are key to understand soft matter behaviour. T h ese basic uni ts are neither macroscopic nor microscopic (in the atomic sense), b ut rather mesoscopic . The relevant scales range from several nanometres to a micrometer. An example of these l ength scales is those associated with (topological) defects in liquid crystal s. Another exam ple is the persistence length as sociated with polymers o rather than t h e shorter scale of atom s and monomer units. Solutions of amphiphilic molecules form nanostructures such as mic elles, vesicles, and tubules. Modularity at the mesos cale is key to theory development. 3. Slow dynamic s The m esoscopic length scales and complex structures lead t o phenomena occurring on time scale s of the order of seconds or minutes. 4. Universali ty The sam e physical properties can arise from materials w ith qui te different underlying chem istries. S oft matte r theories that describe liquid crystals, polymers, coll o ids, and gels a re indepe ndent of che mical detail s and so descr ibe a wide range of mate rials. 5. Common e xperimental techniques The dom inant tools are microscopy, scattering (light, x -rays, neutrons), and rheom eters which mea sure mechanical properties such as viscosity (rheology). These techni qu es matc h the releva n t time and distance sc ale s. Th ese are typically much larger than those associa ted with hard ma terials and so require complementary techniques and instrumentation. 6. Multi-disci plinarity Soft mat ter is studied by physicists, chemists, engine ers, and biol ogists. Qualitat ive difference New states of ma tter are seen such as the sponge ph as e of mi croemulsions, ferroe lect r ic smect ics . Solutions of am ph iphilic molecules form a rich arra y of phases through s elf- assembly, reflecting orde ring of underlying micelles, vesicles, or tubu les. Discontinui ties are se en in phase t ransitions including polymer and gel collapse. o It quant ifies the bending stiffness of a polym er molecule. For parts of the polymer shorter than t he persistence length, the molecule behaves like a rigi d rod. For parts that are muc h longer t han the persistence length the polymer dynamics can be described in terms of a three- dime nsional random walk. Eff ective interactions describe i n teractions between modular structures. Volume exclusion lea ds to entropic attraction between large objects . Entropic elasticity, with a magnitude proportional to temperature, i s rel evant to rubber and polymers. The hydrophobic interaction drives self-assembl y in aqueous solutions of amphiphilic molecules. Exam ples of effective theories include the freely-jointed chain for polymers and the Maier- Saupe theory for liquid c ryst als. A study of the turbule n ce of the fluid fl ow in a system of s wimmi ng bacteria is i llustrative. Wensink e t al. 193 found that a qualitative and quantitative d escription of observat ions of flow patt erns, energy spectra, and vel o city structure functions w as given by a toy m odel of self- propelle d rods (similar to tha t proposed by for flocking) and a mi n imal continuum model for incom pressible flow. For the toy model they presented a phas e diagram as a function of the volume fra cti on of the fluid occupied by rods and as pect ra tio of t h e rods. There were six distinc t phases : dilute state, jamming, swarming, bi onematic, turbu lent, and lane d. The turbule nt state occurs for high filling fractions and in term ediate aspect ratios, covering typical value s for bacteria. Regardi ng th is work, McLeish highlighted the importance of the identification of the relevant mesoscopic scale and the power of toy m od els and effecti ve theories in the following beaut iful commentary. 194 “Individual ‘bacte ria’ are represented in this simulation by s impl e rod-like structures that possess just the two propertie s of mu tual repulsion, and the e xertion of a constant swimming force al ong thei r own length. The rest is simply calculation of the consequences. No more deta iled account than this is taken of t h e com p lexities within a bacte rium. It is somewhat astonishing t hat a model of the intermediate elemental structures, on s uch parsimoni ous lines, is able to reproduce the complex features of the emer gent flow structure. Impossible to deduce inductively the salient features of the underlying physics from the fluid flow alone — creative imagination and a theoretical sc alpel are required: the first to create a sufficient model of re ality at the underl y ing and unsee n sc ale; the second to whittle away at its rough and over-ornate edge s until what is left is the streamlined and necessary model . To ‘understand’ t he turbulent fluid is to have i d entified the scale and struct ure of its origins. To look too cl ose ly is to be confused with unnecessary smal l detail, too coarsely and there is simply a n echo of unexplained pa tte rns. ” 9.4 Spin glasses These are alloys of a non-magnetic metal (e.g., Cu) and a magnetic atom (e.g., Cr). They can be vie wed as a large c on centration of magnetic impurity atoms located in random positions inside t he host metal. The magnetic moments of t wo impurities, separated by a distance R, inte ra ct wit h on e anothe r v ia t h e RKKY interaction, which has a ma gn itude that varies as 1/R 3 and a sign t h a t rap idly oscillates as a func tion of R. Given the random location of the spins this means t hat the interspin inte r actions are random and ca n be of either sign. In different words, this is a system with quen ched disorder and frustrated inte r actions. Novel ty For im pur ity c on centrations larger tha n a critical value, as the temperature i s lowered the mat erial undergoes a transition to a d istinct state of matter , below a low temperature T sg . There i s no long-r ange ma gn etic order or net magnetisation. In this state the direction of the loca l magnetic moments are frozen, albeit i n r andom directions. It is not in an equilibrium state and exhibits relaxation of magnetic properties on macroscopic time scales, i.e ., glass y behavi our. Th is mea ns th ere are long-lived metastabl e stat es. Discontinui ties The t emperature dependence of thermodynamic prop ertie s are qualitatively different from regular magne tic phase transitions. The specific heat capacity is a continuous function of tem perature, with a peak around the temperature T sg, that has a magnitude proportional to the conce ntration of magnetic impurities. At extremely low magnetic fields (of the order of 10 -4 Tesla ) the magnetic susceptibility has a sharp cusp at T sg, which becomes a broad peak for larger fiel ds. This nonlinearity has been interpreted as a singulari ty in the third -order magne tic susceptibility. Discovery of the spin glass state ra ised two fundamental que st ions about t h is state of matter. What i s th e order parameter? What sym metry is brok en? Foll owing the proposal of the toy model s discussed below, th ese answers were answered de finitively by Parisi 195 and highlight how dramat ically different this state of matter is. Toy models The Edwards-Anderson model is a Ising model wher e the i n ter-spin interactions J ij have random val u es assigned by a Gaussian probability distribution with average zero. A simila r but sl ightl y d ifferent model that is more amenable t o analytic tre atment is the Sherrington-Kirkpat r ick model. In it the i n ter-spin interactions are infinite-range and the mean -fi eld theory is exa ct. These toy models capture essentia l ingredients to understanding spin glasses incl uding the rugged energy landscape, r eplica symmetry breaki ng, and ultramet ricity. Repl ica symm etry break ing Theoretic al calculations of the properties of disordered sys tems ca n be performed by us ing the replic a trick which is based on the mat h ematical identit y.            One first conside rs n identical replicas of the system, calculates the pa rtition function Z and then t akes the limit of n tending to zero, making use of the mathematical identity. The t oy models exhibit mu ltiple local equilibria that become infinite in number in the therm odynamic limit. Furthe r more, t he magnitude of the typical energy barriers bet ween loca l minima scales with N. This leads to the concept of a rugged e n ergy landsca p e that can describe th e glassy beha v iour. The e n ergy minima exhibit a h ierarchical struct ur e that can be describe d in terms of th e mat h ematical concept of ultrametricity. In the spin gl ass state, the br eaking of replica symmetry is related t o the existence of many pure therm odyn amic stat es (i. e., local minima). The order parameter and breaking of replica symmet ry can be understood as follows. A repli ca  can be view ed as one realization of the system.   is a measure of the average overlap of the two s olutions  a nd  . The order paramete r is a function q(x) that is def ined i n terms of a probability distribution      and varie s between 0 and 1. q(x) is non-uniform in the glassy phase. Inter-disciplinarit y The underl y ing models, physical ideas and mat hematical techniques are relevant and useful for a wide range of complex systems. 196 This le d to new insights into problems in neuroscie nce, the folding of proteins, 197 biological evolution, computer science, and optim ization in applied mathematics. Examples of th e latte r in clude the tra v elling salesman problem a nd si mulated a nn eali ng. I now discuss how spin glass ideas inspired Hopfield ’ s work on associati ve memory and neural networks. 10. Nuclear physics Atomic nuclei are complex quantum many-body s ystems. The table b elow summa r ises how nucle ar physics is concerned with phenomena that occur at a r ange of le ng th and number scale s. A s the atomic number A, number of protons Z, and number of neutrons (A-Z) in a nucle us increases qualitativel y d ifferent behaviour occurs. The nature of the ground state and low-lying excitations c hanges. Large nucle i exhibit collective ex citati ons. 198 Nuclear electric quadrupole moments are someti mes more than an order of magnitude larger than can be attributed t o a single proton . Rotat ional band structures refl ect the possibility of nuclea r shap es including prolate, oblate, and tri axial d eformations from a sphere. The moments of inertia associated with the rotational state s are markedly smaller t han the moments for rigid rotation expected for uncorrelated single-particle motion. Scale , Number of nucle ons Enti ties Effective inte ra cti ons Effective Theory Phenomena 10 3 - 10 54 Nuclea r matter Fe rmi degene ra cy pressure Equat ion of state Neutron stars, superfluidi ty, quark deconfi nement transit ion 100 Heavy nucl ei Surface t ension Liqui d drop model Fi ssion, fusion, shape vibra tions 8-300  -structure s  -  Cluster m odels  -deca y, nucle osynthesis 4-300 Nucleon pa irs Interacti ng boson model Shape phase transit ions 3-300 Nucleus, nucle on quasipart icles Nuclea r mean- fiel d Shell m odel Magic num bers 1-300 Nucleons, neutri nos Wea k nuclear force Fe rmi V - A theory Beta decay 2 Protons, Neutrons, Pi ons Strong nucle ar force, Pion excha nge Yukawa, Chiral fiel d theory 1 Quarks, Gluons Gl uon exchange QCD Quark confine ment, Asymptotic freedom Tabl e 2. Hi erarchies in nuc lear physics. Moving fro m the bottom leve l to the second top leve l, relevant length scales increase from less than a femtometre to several f emtomet r es. Tabl e entries are not exhaustive but examples. At eac h level of the hierarchy there are effective interacti ons th at a r e describe d by effective theori es. Some big questions in the field concern how the effe ctive theories that operate at eac h level are related to the levels above and below. The be st-known are the shel l model, the Bohr -Mottelson the ory of non-spher ical nuclei, and the li quid drop model. Here I also describe th e Interacti ng Boson Model (IBM), which generalises and provides a microscopic basis for the Bohr-Mottelson theory. Othe r effective theories include chiral perturbation theory, Weinbe rg 's theory for nuc leon -pion interactions, and Wigner's random matri x theory. The c hallenge in the 1950 ’ s was to reconcile the liquid drop model and the nuclea r she ll model . This led to discovery of collective rotations and s hape deform ations. S ince the 1980s a ma jor challenge is to show how the strong nucl ear force betwee n two nucleons can be derive d from Quantum Chromodynamics (QCD). Fi gure 6 ill ustr ate s how the attracti v e inte ra cti on b etween a n eutron and a proton can be understood in te r ms of creation and destruct ion of a down quark -antiquark pair. This program has had limited succe ss , e.g., in cal culating from QCD the strengt h of the three-nucleon interaction or the parameters in chiral effective theory. Fi gure 6. Effective interacti ons in nu clear physics. 199 Proton-neutron sca ttering can be describe d in terms of virtual exchange of a pion or in terms of creati on and annihi lation of quarks describe d by Quantum ChromoDynamics (QCD). An outstanding probl em concerns the equation of state for nuclear m atter, such as found in neutron sta rs. A challenge is to learn more about this from the neutron star mergers that are dete cted in gravitational wave astronomy. 200 Eff ective theories The c entrality of emergence in nuclear physics can be illustrated by considering the diffe r ent effective theories that have been proposed to describe phenomena at different sc ales. Liquid drop model A nucleus is tre ated as a drop of in compre ssibl e liquid with an ene rgy E(A,Z). This ene rgy function ha s separate terms corresponding to the vol ume, surfac e, C ou lomb i n teraction, nucle on asymmetry, and pairing. Parameters in the model are determi ned by fitting the observed binding e n ergies of a wi de range of nuc lei. The theory can be use d to explain spontaneous fission and e stimate the energy barrier to it. Shell model The shel l model has s imilari ties to microscopic mod els in a tomic physics. A maj or achi evement is it explains the origins of magic numbers, i.e., that nuclei with atomic numbers 2, 8, 20, 28, 50, 82, and 126 are pa r tic u larly sta b le because t hey have closed shells. Other nucle i can then be described theoretically as an inert closed shell plus vale n ce nucleons that inte ra ct wit h a mea n-f ield potential due to the core nuclei and then with one another via effective interactions. Landa u’s Fermi liquid theory provides a basis for the nuclear shell model which a ssum es th at nucle ons can be described in terms of weakly interacting quasiparticles moving in an average potent ial from the other nucleons. 201 The BCS theory of superconduct ivity, was adapted to describe the pairing of nucleons, leading to energy di fference s between nuclei with odd and even num bers of nucleons. This helped explain why the moments of ine r tia for rotati on al bands were smal ler than for independent nucleons. The shell model is the basis of many-body cal culations that include interactions between nucleons . This means conside r ing Slater dete rminants (configurations) which d escribe d ifferent occupations of the orb itals associat ed with the nuclea r mean-field potential. A me asure of the complexit y of this many-body system is that a J=2+ state of 154 Sm can contain contribution s from more than 10 14 configurations involvi ng just a single nu cleon moved to an exc ited orbital . This highlights the necessity of simpler t h eorie s such as those considere d below and makes their succe sses even m or e impressive . Cluster structures For ce r tai n nu clei there are hints in s hell model calculations of internal structures that consis t of alpha pa rticles. These are particularly evident in e xcit ed states near the threshold for alpha parti cle emission. The excited state of t he second J=0 + state of 12 C has a structure of three alpha particles. This state was made famous by Fred H oyle a s the fine - tuning of its’ ene rgy is nece ssary for the stellar nucleosynthesis of carbon. T his internal struct ur e is consistent with trends seen i n binding energies . These hi n ts are also seen in a t oy model wher e all the nucle ons move in a pot ential which is an ani sotropic harmonic oscillator. T h is has motivated effective theories where a lpha pa r ticles are qua s iparticle s . 202 Isolate d nuc lei reside ne ar a quantum phase transition between a nuclea r liquid and a Bose- Einste in condensate of alpha clusters. 203 In theoretical models, changing the range and loca lity of th e nuclear forc e, one can traverse from o ne phase to the othe r. In d ifferent words, if the density of everyd ay nuclear matte r is reduced by a fact or of three, t he nucleons condensed i n alpha particles. Bohr-Mottel son model This model br idges the liquid drop and shell models and descri bes collective effects in hea vy nucle i such as shape vibrations and la rg e electric quadrupole moments. The effective Hamil tonian can be written in terms of the k inetic e n ergy and pote n tial ene rgy associated with two nuclear deform ation parameters  and  . Th e first te r m of the Ha miltonian is the kinet ic energy of the  -v ibrati ons. Th e second term describes the rotational energy and the kinet ic energy of the  -vibra tions and contains the C asimir ope r ator of the SO(5) Lie group. The l ast term is the potential energy as a function of both deformation parameters  and  . Interact ing Boson Model (IBM) The IBM explains the ra p id change s as a fun cti on of the numbe r of nu cleons observed in the low-lying excitation spe ctrum of nuclei contain ing even numbers of protons and of neutrons. R 4/2 is the ra tio of the energies of the J=4 + state to that of the 2 + state, relative to the ground state . As the system changes from that of nucleus with close to a magic number of nuclei to an al most half-fi lled shell and elli psoid al shape, R 4/2 change s from l ess than 2 to about 3.3. In betwee n, th e ratio is 2.0 and t he nucleus exh ibits spherical vibrations. Ass ocia ted with this rapid cha ng e is a large i ncrease in B(E2 ), the str ength of the quadrupole tra nsi tion between the 2 + state and the ground state. The IBM is surprisingly sim ple and successful. It is an exam p le of the power of toy models and effecti ve theories. It ill ustr ates the im por tance of quasiparticles, builds on the stability of closed shel ls and neglects many degrees of freedom. It describes even -even nuclei, i.e., nuclei with an e ven number of protons and an even number of neutrons. The ba sic entities in the theory are p airs of nucl eons, w hich are either in an s- wave or a d-wave state. T here are five d- wave stat es (corresponding to the 2J+1 poss ible states of total angular momentum with J=2). Eac h state is represented by a boson creation operato r and so the Hil bert space is six - dime nsional. If the states are degenerate [which they are not] t h e model has U(6) symmetry. The IBM ca n describe transitions in the shape of nuclei 204 and can be justified from the shell model with certain assumptions. 205 The IBM Hamiltonian is written i n terms of the most general possible combinations of the boson operators. This has a surprisingly simple form, involvi ng only four p aramete rs,       󰇍    󰇍    󰆒  󰇍    󰇍    󰆒󰆒  󰇍    󰇍  , where   is the num b er of d-bosons,  󰇍  is the boson angular momentum op erator, and  󰇍  and  󰇍  roughly correspond t o quadrupo le and pairing i n teractions be tween bosons , respecti vely. For a gi v en nucleus the four parameters can be f ixed from experiment , and in principle cal culated from the shell model. The Hamiltonian can be writte n in a form that gives physica l insight, c onnects to oth er nuclear models and is amenable to a group theoretical a n alysis that make s calculation and understanding of the energy spectrum r elatively simple. Central to the group theoretical analysis are subalgebra chains which connect U(6) to O(3), rotat ional symmetry. Th e three d istinct chains of broken symmet ry are associated with t he subalgebra s U(5), S U(3), and O( 6 ). The forme r two connect to an anh armoni c vibrator and to a quadrupol e deformed rotator, respe ctively. This provides an elegant connection to Bohr and Mottel son ’s model of collective quadrupole surface excitati ons and Elliott’s SU(3) model. For each of t he three subalgebras the Ha miltonia n can be written in terms of Casimi r operators of t he corresponding subalge br a and so are analytic ally soluble. A major a chievement of the model, and which was unant icipated, was that the O(6) limit provided a qualitati v e and qua ntitative description of a doze n of the lowest-lying excitations of the 196 Pb nucleus. 206 It described the ordering and spacing of the levels, thei r quantum numbers, and the intensity of gamma ra y transitions between t hem. For all e ven- even nuclei i n the nucleotide chart estimates of the parameter values for the IBM model ha v e bee n made. The different parameter regimes of the model c an be related to the different suba lgebra chains. Nuclea r theorists have suggested that the model a nd the associated nuclei exhibit quantum phase tra nsitions as the model parameters and nuclei numbers change. 204 However, these are crossovers not true pha se transitions. The nu cle us ha s a finite nu mber of degrees of freedom and the re is no spontaneous symmetry brea king : the ground state has J=0. On the other hand, the l ow-lying excitation spect ru m does reflect the cross over. Perhaps, this is anal ogous to the “tower of sta tes” first introduced by Anderson to describe how a n an tiferromagnetic Heisenberg m odel on a fin ite lattice exhibits signatures of incipient spont aneous s ymmet ry breaking whic h is prevent ed by quantum tunneling between degenerate ground states. 207,208 Taki ng the limit of an infinite number of bosons (nucleon pa irs) giv es a semi-classical model, all owing connection with the Bohr -Mottelson model. For example, in the SU(3) case, taking the l imit where the qu antum numbers  ,  become large the y tend to the nuclear deformati on paramete rs  and  th at appear the Bohr-Mot telson model. Random matrix the ory Wigne r proposed th is could describe the stati st ical distribution of energy le v el spaci ngs in heavy nuc lei. Th is effective t h eory makes no assumptions about t h e detail s of in teractions betwee n nucleons, except that the Hamiltonian matri x has unitary symmetry. Universality is evide nt in that it can describe experimental data for a w ide range of heavy nuclei. Furt hermore, rando m matrix the ory can also describe aspects of quantum chaos and zeros of the z eta function relevant to number theory. It h as also been applied to ecology a nd finance, as discussed lat er. Skyrmion model Skyrme conside red a classical field theory defined in terms of an SU(2) matrix describing three pi on f iel ds and dyn amic s def ined by a non -linear sigma model Lagrangian. 209 So lutions with topol ogical charge A were identified with a nucleus with atomic number A. It is remarka b le t h at nucleons eme rg e purely from a pion fiel d theory. Witten showed that Skyrme’s model could be regarded as the low -energy effective field the ory of Quan tum ChromoDynami cs (QCD ) in the limit of a large number of quark c olours. 210 Chiral ef fective field theory Wei nberg proposed a theory to describe the interaction of nucleons and mesons 211 . It reproduced re sul ts from curre nt algebra calculations. Chiral perturbation theory is a systemat ic derivation of Weinberg’s theory that ensures the theory is consistent with the symmet ries of Q CD. 212 As discussed below, developing t he theory led Weinberg to change his views about e ffective theories and renormalizability as a criteria for theory selection. 211 11. Elementary particle s and fields The system s of interest can be viewed as being compos ed of many i nteracting components beca use quantum fields have components with a continuous range of momenta . Even the “vac uum” is composed of many fluctuating fields, and an elementary particle interacts wit h these fl uctuating fields. Elementa ry p article s c an be viewed as excitations of the vacuum, just as in quantum condensed matter quasipa rticles can b e viewed as elementary excitati ons of th e ground state . Big que stions Big questi ons in the fie ld include unification, quantum gravit y, the hierarchy problem, baryon asymme try, and connections to big -bang cosmol ogy, includi ng the origins of dark matte r and dark energy, a nd the val u e of the cosmological constant. The hie r archy problem is t hat measured values of some masses and c oup ling constants in th e Standard Model are many orders of magnitude different from the "bare" values used in the Lagrangian. F or example, why is the weak force 10 24 tim es stronger than gravity? In different words, why is the Higgs mass parame ter 10 16 times smaller tha n the Pla n ck mass? There a r e two aspec ts to the problem of th e cosmological constant. Fi rst, the measured value is so small, 120 orde rs of magnitude smaller than estimates based on the qu antum v acuum energy. Second, t h e mea sured v alue seems to be f inely tuned (to 120 significant fi gur es!) to the va lue of the mass energy. Scale s and stratum Scale , Energy Enti ties Effective inte ra cti ons Effective Theory Phenomena Ele ctrons, photons Coulomb Quantum ele ctrodynamics (QED) Vacuum polari sation, e-e+ creation, anoma lous magne tic moments 1 MeV Nucleons and pions Strong nucle ar force Chiral EF T 200 MeV Light quarks, gluons Heavy quark EF T Quark confine ment, ch iral symmet ry breaking 2 Ge V Heavy quarks QCD 100 Ge V W and Z ga ug e bosons Wea k nuclear force Wei nberg- Salam Beta decay, Neutral currents 125 Ge V Higgs boson, lept ons Standard m odel Electrowea k symmet ry breaking, lepton mass 10 16 Ge V SU(5) GUT Proton deca y 10 19 Ge V Strings, space- tim e foam Quantum gravity Hawking radia tion Tabl e 3. The energy scal es and ass ociated stra tum as sociated with eleme n tary particles and fiel ds. E FT i s Eff ective Field Theory. As far as is known there is no qualita tively new phys ics betwe en 100 GeV and 10 16 Ge V, a range cove r ing 14 orders of magnitude, a nd b eyond the range of curre n t experiments. This absence is some times referred t o as the “desert”. Novel ty At eac h of the stratum show n in Table 3 n ew entities emerge and there are effective theories describi ng their interactions. A s in condensed matter physics, a unifying concept is spontaneous symmetry breaki ng , introduced by Nambu, drawing on anal ogu es with superconduct ivity. 213 It leads to new particles and effect ive interactions. Some massless parti cles acqu ire ma ss. For example, the W and Z gauge bosons in the electroweak theory. When the r eleva nt quantum field theories are consid ered at non- zero temperature spontaneous symme try breaking is associate d wi th phase transitions at temperatures that would have oc curred in the very early universe. 1212214 218214 For exampl e, electroweak symmet ry breaking occurs at a temperature of about 10 15 K, which was the te mperature of the universe about 10 -12 sec after the big bang. Anomalie s refer to where a qua n tum field t h eory does not have the same symmetry as the cla ssical Lagrangian on which the theory is based. A famous example is the Adler – Bell – Jackiw anom aly, w hich explains the d eca y r ate of neutral mesons , and describes the non- conservat ion of axial currents. Th ey are conserved i n the classical Lagrangian but not in the quantum theory. Dimensional transmutation concerns how the quant u m fluc tuations associated with inte ra cti ons c an lea d to the emergence of energy scal es. For example, consider a classical fiel d theory in which the coupling consta n t describing the interactions is a dimensionless constant . In th e correspondi ng quantum field theory , there may be l og arithmic divergences in one-loop dia gr ams of perturbation t heory . This implies that this strength of the interactions depends on  , the typica l energy scale of the properties of inte r est. From a renormalization group perspecti v e thi s is referred to as the "running" of the coupling constant and is described by the be ta function.  is sometimes refe rr ed to as the cut off (range of validity) for the perturbation the ory. Depending on the sign of the bet a functi on,  may be a high energy (UV) or low energy (IR) scale. Note t h at  is not predicted by the theory but is a parameter that must be fixed by experiment, and it specifie s th e bounds of validit y of the t h eory. Universalit y is reflecte d in the theory being valid over a wide range of energy scales and the beta func tion de f ining the relationship between t h e corresponding theories. Modularity at the mesoscale Composite par ticles such as nucleons and mesons can be viewed as elem entary ex citations of the va cuum for Quantum ChromoDynamics (QCD ). These particles interact wit h on e another via e ffe cti v e inte r actions that are much weaker than the inte ractions in the underlying theory (Fi gure 6). Spe cifically, the force bet we en nucleons is s hort range a nd much weaker than the force bet ween the quarks of which they are composed. The composite particles have masses (energie s) intermediate between the IR and UV cutoffs of the theory. In that sense they occur at t he mesoscale. In some the ories there are s olit on or instanton solutions the classical e qu ations of motion. Colem an referred to these as “classical lumps” , 215 and identified their quantum descendants with elementary partic les. In the QCD vacuum t h ere are centre vortices in the SU(3) gauge fiel d. These are thick two- dime nsional objects imbedded in four-d imensional space time. They determi n e the properties of quark confinement and dynamical symmetry breaking. It has become possible to visualise cent re vor tices in rece n t lattice QCD simulations. 216 Eff ective theories Over the pa st fif ty years the re has been a significant s hift i n perspective concerning effective theori es. Originally, QED, th e Standard Model, and quantum field theory in general, were considere d to b e fundame n tal . R enormalisability of a theory was considere d to be a c r iteria for the validi ty of that theory, arguab ly followi ng the work of 't Hooft a nd Veltman showing that Yang-Mil ls theory was renorm alisable . In 2016, Steven W einberg described how his perspecti v e cha nged, following his development in 1979 of chiral effective theory for nucle ons. 211 “ Non-renormaliza ble theories, I realized, are just as r enormaliza ble as renormalizable theori es … For me in 1979, the answer involve d a ra dical reconside ration of the nature of quantum field theory … P erhaps the most important less on from chira l dynamics w as that we should keep a n open mind about renormalizability. The renormalizable Standard Model of ele mentary particles may itself be just the first term in an effective field theory that contains every possible i n teracti on allowed by Lorentz invariance and t he SU (3) × SU (2) × U (1) gauge symm etry, only with the non-renorm alizable terms suppressed by negat ive powers of some very l arge mass M... confirmation of ne u trino oscillations le nds support to the view of the Sta ndard Model as an effective field theory, with M somewhere in t he neighborhood of 10 16 Ge V. The Standa rd Mode l is now considered a n effective f ield t h eory. Effective theories have been reviewe d by Georgi 217 and Levi. 69 The associ ate d h istory and phil osophy has be en reviewed by Hartma nn 218 and by Crowther. 219 There a r e two disti n ct approaches t o finding effective theories at a particular scale, re f erred to as bottom-up a nd top-down approache s. H owever, as noted earlier these terms are used in the opposite sense t o in condensed matter physics, chemistry, and biol ogy. To avoid confusion , I will he re refer to them as low-en ergy and high-energy approaches. High-energy approa ches require having a theory at a higher energy scale a nd then integrating out the high energy degrees of freedom (fields and particles) to find an effective theory for the l ower energy scale. This is what Wilson did in his approach t o critical phenomena. The renormalisat ion group provides a technique to compute the effective interactions at lower energi es. In quan tum field the ory a successful example of the high-energy approach is the ele ctroweak theory of Weinberg and Salam. In th e low energy limit, it produced quantum ele ctrodynamics and the theory of t h e nuclear weak interactions. Low-energy approac hes c an be done wi thout knowing the higher energy theory. Sometimes the L agrangian for the effe cti v e theory can be written down based on symme try considera tions and phenomenology. An example is Fermi's theory of beta decay and the weak inte ra cti ons. As for condensed mat ter, the his torical progression of theory development has always be en from low-energy t h eorie s to h igh-energy theories. In different words, the low-energy theory has been de veloped, understood, and tested first. La ter, high-energy theories have been devel oped that have been shown to reduce to the low -energy theory in that limit. This requirement ha s guided the development of the high- energy theory. Toy models Toy mode ls have played a central role in understanding quantum field t h eory , as these model s s how what is possible and aid the development of concepts. When first proposed it was known that many of t h ese toy models did not des cribe known physic al forces or parti cles. M any are not in the 3 +1 dimensions of the real world. Sidney Cole man was a master at using toy models to illuminate ke y aspects of quantum field theory. 215 He discussed models in 1+1 dimensions in cluding t h e sine-Gordon, Gross-Neveu, and Thi rring models. Th ey illustrate how theories with simple Lagrangians can ha v e ric h propertie s in cluding spontaneous symmetry breaki ng , singularities (n on-perturbative behavi our), and a rich pa r ticle spectrum. The Gross- Neveu model exhibits asymptotic freedom, sponta n eous symmet ry breaking, and d imensional transm u tation. The Sine-Gordon model ha s a ground state and excitation spect ru m that changes qualitatively a s the coupling constant in crease s. The classical t h eory exhibits solitons and there are analogous entities in the qua ntum theory. Al though the theory is bosonic these excitations can have f ermionic characte r. I now consider a few ot her examples of toy models: the Anderson model for massive gauge bosons, the decoupl ing theorem, scalar electrodynam ics, and l attice gauge theory. The Nambu-Jona-Lasinio model Nambu showed the im portance of spontaneously broken symmet r ies in quantum field theory, guided by a n analogy with the theory of superconductivity. Following the work of Bogoliubov and Anderson, he showed in 1960 that in the BCS theory “ g auge invariance, the energy gap, a nd the c ollective excitations are logically related to each other ” and th at in case of the strong interactions “ we ha ve only to replace th em by (chi r al)  5 invariance , b aryon mass, and the mesons . ” 213 This connecti on was w orked out explicitly i n two papers in 1961, writte n with Jona- Lasinio. In the first pa p er, 174 they acknowledged, “ that the m od el treated here is not re ali st ic enough to be com p ared wit h the actual nucleon problem . Our purpose was to show that a new possibility exists for field theory to be richer and more c o mplex than has been hi therto envisaged, …” The m odel consists of a mass less fermi on field with a quartic interaction that has chi r al invari ance, i.e., unchanged by global gauge transformations a ssoci ated with the  5 matrix. At the m ean-fie ld level, t his sym metry i s broken. Excitations incl ud e massle ss bos ons (associat ed with the symmetry breaking and like those found earlier by Goldstone ) and bound fermion pairs. It was conj ectured that these exc itations could be analogues of mesons and baryons, respect ively. The model was proposed before quarks and QCD. Now, the fermion degrees of free dom would be identified with quarks, and the model illustrates the dynamical generation of quark m asses. When generalised to include SU(2) or SU (3) symmetry t he model is considered to be an effective field theory for QCD , 220 simil ar to the chiral effective theory used in nuclea r physics. Anderson’s model for mass ive gauge bosons In 1963 Anderson argued that the ca se of supercond uctivity was relevant to how m assless gauge bosons could a cquire mass. 221 It is a toy model in that it is not Lorentz i nv ariant or a non-abelian gauge theory. In a neutral superfluid the U(1) gauge symme try is broken and there a r e associa ted massless collective modes , and they are density oscillations. Bogol iubov had shown that the se modes can also ensure that phys ica l quantities are gauge invariant. In a superconduct or, which is a charge d superf luid, the Coulomb interaction (which is itself medi ated by mass less gauge bosons, i.e., phot ons) causes the density oscillations to ha v e mass (i.e., non-ze ro frequency at long wavelengths). These are plasmons. Anderson’s toy model provi d ed justi f ication for the belief that symmetry bre aking could lead to gauge bosons acqui ring mass , which turned out to be the c ase for t he non-A belia n g auge the ory for the ele ctroweak interaction. Anderson’s w ork inspired Higgs work on the Higgs field a nd its associat ed boson. Scalar el ectrodynamics This consists of a sca lar massless boson field (a meson) interacting with an Abelian gauge fiel d [i.e., electromagnetism]. It was studied by Sidney Coleman and E r ik Wei nberg in 1973. 222 They showed how radia tive corrections beyond the semi- cla ssic al approximation lea d to spontaneous symmetry breaking. In different words, the inte ractions (and associated quantum fluctuations) lead to the sys tem having qualitatively different properties. In the broken symme try state the excitations (effective fields ) are a m assive vector boson and a massive sca lar boson. The model also exhibited dime nsional transmutation . In fact, this paper is the ori gin of that term. This work w ent a gainst the view of quantum field theory only being perturbative and promoted the importance of effective field theory. It illustrate s vio lati on of adia batic continuity, or in different words, non-pertu rbative b ehaviour. The e xcitations (fiel ds) in the effective theory are not the same as thos e in the original classical Lagrangian. They a lso extended their analysis to non-Abel ian gauge the ories. The decoupl ing theorem In 1975, Appelquist a nd Carazzone 69 considered a set of massless gauge fields coupled to a set of ma ssive spin-1/2 f ields, and showed that the lo w-energy theory was sim ply a gauge theory wit h renor malised interac tions. Thus, the gauge fields decouple d fro m the massive fiel ds. They called t h is the decoupling theorem and argued it was true for a wide range of theori es. Th is conce p t is central to the emergence of effective field theorie s (EFTs) and a hierarc hy of scales, as occurs in unified theories. 217 In its simpl est case, this theorem demonstra tes that for two coupled systems with different energy scales m 1 and m 2 (with m 2 > m 1 ) and descri b ed by a renormali sab le theory, the r e is always a renormalisation condi tion acc ording to which the effects of the physics at scale m 2 can be effectivel y included in the theory wit h the smaller scale m1 by changing the parameters of the corre sponding theory. The de coupling theorem implies the existence of an EFT at scale m 1 . However, the EFT will cea se to be applicable once the energy gets close to m 2 . Unific ation and GUT s In 1974, Ge orgi and Glashow proposed a Grand Unified T heory ( GUT) to unify ele ctromagnetism with the weak and strong nuclear f orces. The symmetry group SU(5) can be broken down to SU(2) x U(1) x SU (3) where the thre e subgroups correspond to the w eak, ele ctromagnetic, and strong nuclear force respectivel y. This symme try breaking can occur through the Higgs mechanism acting at a h igh energy scale of order 10 17 GeV. The decoupl ing theorem can be used to s how that the strong interaction decouples from the ele ctroweak interaction and is describe d by QCD. Below this high energy scale there are effective ly three independent coupling constants , corresponding to the three forces, and obeying t heir own s cali ng r elations. At la boratory sc ales they differ by many orders of magni tude but become comparable a t the un ification scale. In different words, the standard model emerges as a low -energy effective theory following symmetry breaking. Although this GUT has ae sthetic appeal it is unlikely to be correct as it predicts the existence of magnetic monopole s and th e spontane ous de cay of protons. Neither have be en observed. The pro ton's half-li f e is constrained t o be at least 1.67×10 34 years, w hereas most pre dictions of the GUT give a value that is one to three ord ers of magnitude smaller. Lattic e gaug e the ory Lat tice gauge theory was arguably a toy model when first proposed by Wilson in 1974. He treated space -time as a discrete lattice, purely to make analysis more tractable. Borrowing insights and t echniques from lattice model s in s tatistical mechanics, Wilson then argued for quark confineme nt, showing that the confini ng po tential was linear with dista n ce. Earl ier, in 1971 W egner had proposed a Z 2 gauge theory in the context of generalised Ising model s in sta tistical mechanic s to s how how a phase transiti on was possible without a local order paramete r, i.e., without symmetry breaking. Later, it was shown that th is phase transit ion is s imilar to t h e confi n ement-deconfinement phase transition t h at occurs in QCD. 223 In condensed m atter, Wegner’s w ork provided a toy model to illustrate the possibility of a quantum spin liquid. 224 Lattic e QCD The di screte nature of lattice gauge theory means it is amenable to num erical simulation. It is not nec essary to h ave the continuum limit of real spacetime because of universalit y. Due to increa ses in computational power ove r the past fif ty years and innovations i n algorithms, lat tice QCD can be used to calculate properties of nucleons and mesons, such as mass and deca y ra tes, with impressive accuracy. The mass of three mesons is typically used to fix the mass of the l ight and strang e quarks, and the length scale. The mass of nine other particles, incl uding, the nucleon, is calcu late d wi th an uncertainty of less than one per cent, and in agreement with experimental values. 225,226 An indication t hat this is a strong coupling problem is that abou t 95 per ce n t of the mass of nucleons comes from the interactions. Only about 5 pe r c ent i s from the rest mass of the constituent quarks. Like the Ising model, lattice QCD s hows how because of uni versality toy models can some times even give an excellent descript ion of experimental data. Phase diagram of QCD QCD has a rich phase dia gram as a func tion of temperature versus baryon density. 227 This spans the range from the “l ow” d ensities and temperature of everyday nuclear matter to neutron sta rs, and to the high temperature s of th e early universe and relativistic heavy ion coll id ers. There is a first-order phase transition between quark confinement and the deconfi nement of the quark-gluon plasma . An open questions is whethe r there is a critical point at low density a nd h igh temperature. 228 Colour superconductivity has been proposed to exist a t high densities and low t emperatures nea r the deconfinement transition. 229 It is possible tha t this transition occurs inside the dense cores of n eutron stars. 230 Bey ond the Standard Model The Standa rd Mode l (SM) is now w idel y considered to be an e ff ective the ory . 231,232 An approach t o f inding physics be yond the SM is to write down a Lagrangian with SM as the lea ding term and next term is in powers of E -6 where E i s an energy scale associated with a higher-ene rgy theory from whic h the SM emerges. T his provides a framework to look for small deviations from the SM at currently accessible experimentally e nergies and to propose new experi ments. 12. Quantum gravity Einste in’s theory of General Relativity s ucce ssfully describes gravity at large scales of le ng th and ma ss. In contrast, quantum theory describes small scales of length and mass. Emergence is cent ral to most attempts to unify the two theories. Crowther has given a comprehe nsive discussion, from a phi losophical perspective, emphasising the central role of effective field theory. 219 Before consi dering specific exa mple s, it is useful to make some distinctions. First, a quantum theory of gravity is not necessarily the same as a theory to unify gravity with t h e three other forces descri bed by the Standard Model. Whe ther the two problem s are inextricable is unknown. Second, there are two distinct poss ibilities on how class ical gravity might emerge from a quantum theory. In Einst ein’s the ory of General Relativity, space-time and gravity are inte rtwined. Cons equently, the two possibilities are as follows. Space-time i s not emerge n t. Classical General Relativity emerges from an und erlyi ng quantum field theory describ ing fiel ds a t small length scale s, probably compara b le t o the Planck l ength. Space-time emerges from some underlying granular s tructure . In some limit, classical gravity eme rges with the space - time continuum. Third, the r e are "bott o m-up" and "top-down" approache s to discovering how classical gravity eme rges from an underlyi ng qu antum theory, as was emphasised by Hu 233 and Crowther. 234 As discussed earlie r abou t conflicti ng terminology, here I will refer to low-en ergy and hi gh- energy approa ches. Fi nally, there is the poss ibility t hat quantum theory itself is emergent, as discussed in the earl ier section on the quantum measurement problem. Some proposals of Emergent Quantum Mechani cs (EQM) attempt to al so in clude gravity. 149 I now menti on several different approaches to quantum gravity and for each point out how they fi t into the distinctions above. U niversality presents a problem for the high-energy approache s high lighting the problem of protectorates . General Relativity has been shown to be the l ow-en ergy limit of seve r al very different underlying theories. Th is shows that just derivi ng Gene r al Relati v ity from a higher energy the ory is not a sufficient condition for finding t he correct quantum theory of gravity. Crowt her 234 stated that , “there is very little that tie s the emergent theory to the one that it emerges from… the high - energy theory is severe ly underdete r mine d. ” Gravitons and semi-classical t h eory A simple low-ene rgy approach i s to s tart with classical Gene ral Relativity and consider gravitati onal waves as the normal modes o f oscillation of the spac e- time conti nuu m. They have a linear dispersion relation and move with the speed of light. They are analogous to sound waves in an el astic medium and electromagnetic wave s in free space. Sem i-classical quanti sation of gravi tational wave s leads to graviton s which are a massless s pin-2 field. T hey are the anal ogue of phonons in a crystal or photons in the electromagnetic vacuum. H owever, this reve als nothing about an underlying quantum theory, just as phonons with a linear dispersion rel ation reveal nothing about the underlying crystal structure. On the othe r hand, one can start with a massless spin-2 quantum field a nd consider how it scatt ers off mass ive partic les. In th e 1960s, Weinber g showed that ga uge invariance of the scatt ering amplitudes implied the equivalence princi ple (inertial and gravita tional mass are ident ical) and the Einstein fie ld equations. In a sense , this i s a high-energy approach, as it is a deriva tion of General Relativity from an underlying quantum theory. In passing, I men tion Wei nberg used a similar approach to derive ch arge conservation a nd Maxwe ll’s equations of cla ssical electromagnetism, and classical Yang-M ills theory for non-abelian gauge fields. 235 Wei nberg pointed out that this cou ld go agai nst h is reductionist claim that in the hierarc hy of the sci ences the arrows of the explanation always point down, s tat ing that “ sometimes it isn't so clea r which way the arrows of explanation point … Which is more fundamental, general relativi ty or the existence of particles of mass zero and spin two? ” 20 More recently, We inberg discuss ed General Relati v ity as an effective field theory 211 ... we should not despai r of applying quantum field theory to gravitation just because the r e is no renormaliz able theory of the metric tensor that is invariant und er general coordi n ate transforma tions. It increasingly seems apparent that the Einstein – Hilbe rt Lagrangian √gR is just the least suppressed term in the Lagrangian of an effective fi eld theory containing every possible gene rally covariant function of the metric and its derivatives... This is a low-ene rgy approach that was first e xp lored by S akharov i n 1967, 236 and experi enced a renewal of interest around the turn of the century. 237 Wei nberg then went on to discuss a high-energy a pproach: “it i s usually assumed that in the quantum theory of gravitation, when Λ reaches some very high ene rgy, of th e order of 10 15 to 10 18 GeV, t he ap propriate degrees of free do m are no longer t he metric and the Standard Model fields, but s omethi ng very different, perhaps strings... But maybe no t..." String the ory Versions of string theory from the 1980s aimed to unify all four forces. They were formulated in te rms of nin e spatial dimensions and a large intern al symmetry group, such as SO(32), where supersymmet r ic strings were the fundamental units. In the low energy limit, vibrations of the stri ngs are identifie d wi th elementary particles in four-dimensional space-time. A parti cle with mass zero and spi n two appears as an immediate consequence of the symmetries of the stri ng theory. H ence, this was origina lly claimed to be a quantum theory of gravity. However, subsequent deve lopments have found that there a r e many alte rn ati v e string theories and it i s not possible to formulate the theory in terms of a unique vacuum. AdS-CFT correspondence In the c ontext of string theory, th is correspondence c onject ures a connection (a du al relation) betwee n a theory of gravity in Anti-deSitter space-time (AdS) and a conformal field theory (CFT ) for a s upersymmetric Yang-Mills gauge t heor y. It is argued t hat it illustrates a holographic pr inci p le for quant u m gravity as the gauge theory is def ined on the boundary of the spac e- time and com p letely determines the theory of gravity. String theorists interpret the duali ty as implying that space- time is eme rg ent. 238,239 Crowther disagre es: 219 “ Even t hough the relationship between the two theories is symmetric, and the duali ty is considered to be exact, many authors take the AdS/CFT dualit y as impl ying that the gauge theory on the boundary is fundamental, while the higher- dime nsional bulk spacetime is emergent from it … The fallacious line of reasoning t hat leads to the suggestion that the gravitational theory is emergent is that an exact formulation of it has not been found, whereas we do have a n exact formula tion of the gauge theory, and so (the thought is that ) the gauge theory is more fundamental . Instead, a n appropriate interpretation of the duality would be one that treats the dual t heories as being on equal footing. Although the duality apparently features two (very) differe nt theories, we should view them as describi ng the same physical quanti ties, just using different concepts … Describ ing one theory as emergent fro m the ot her, on this picture, is misguided. ” In 2018, Witten argue d that AdS-CFT suggests that gauge symmetries are eme rg ent. 240 However, I cannot follow his argument. Seiberg reviewed diffe rent approa ches, within the string theory community, that lead to spacet ime be ing em ergent. 238 An example of a toy m odel is a matrix model for quantum mec hanics (w hich ca n b e vie wed as a zero-dimensional field theory). Perturbation expansions can be viewed as discretised two-dim ensional surfa ces. In a large N l imit, two-dimensional space a nd general covariance (th e starting point for general r elativity) both emerge. Thus, this shows how both two-dimensiona l gravity and spacetime ca n b e emergent. However, this type of emerge n ce i s dist inct from how low-en ergy theori es eme rge. Se iberg also notes that there are no exampl es of toy models where time (whi ch is associated with locality and ca usality) is eme rgen t. Loop quantum gravity This is a high-e n ergy approac h wher e both space-tim e and gravity emerge together from a granular st ruc ture, sometime s ref erred to as "s pin foam" or a “spin ne twork” , and has been reviewe d by Rove lli. 241 The s tarting poin t is Ashtekar’s demonstrati on that General Relativity can be described using the phase space of an SU(2) Yang -M ill s theory. A boundary in four- dime nsional space- time can be d ecomposed into cells and this can be used to define a dual graph (lattice)  . The gra v itational field on this discretised bounda ry is represented by the Hilbert space of a lattice SU(2) Y ang-Mills the ory. T he quantum numbers us ed to define a basis for th is Hilbert space are the graph  , the “ spin ” [SU (2) quantum number] associat ed with the face of each cell, and the vol u mes of the cells. The Planck length limits the size of the c ells. In the limit of the continuum and then of large spin, or visa verse, one obtai ns Ge neral Relativity. Quantum thermodynamics of ev ent horizons A low-energy approac h to qu antum gr avity was taken by Padmanabhan. 242 He emphasi ses Boltz mann's insight: "matter can only store and transfer heat because of internal degrees of freedom". In othe r words, if something has a temperature and entropy then it must have a mic rostructure. He does th is by consideri ng the conn ecti on b etwee n event horizons in Ge neral Relativity and the temperature of the thermal radiation associated with them. He frames hi s research as attempting to estimate Avogadro’s number for space -ti me. The t emperature and entropy ass ociated wit h event horizons has been calculate d for the following speci f ic space-times: For acce lerating frames of reference (Rindler space-time) t h ere is an event horizon whic h exhibi ts Unruh radiation with a temperature that was cal culated by Fulling, Davies and Unruh. The bl ack hole horizon in the Schwarschild metric has the temperature of Hawking radiati on. The c osmological horizon in deSitter space is associated with a temperat ur e proportional to the Hubble constant H, as discuss ed in detai l by Gibbons and Hawking. 243 Padmana bhan cons iders the number of degrees of freedom on the boundary of the event horizon, N s , and in the bulk, N b . He argues for the ho lographic pr inciple that N s = N b. On the boundary surface , there is one degree of freedom ass ociated wit h every Planck area , N s = A/ L p 2 , where L p is the Planck length and A is the surface area, which i s re lated to t h e entropy of the hori zon, as first discussed by Bekenstein and Hawking. In the bulk, classical equipa rtition of energy is assumed so the bulk energy E = N b k T/2. Padmana bhan g ives an alternative p erspective on c os mology through a novel derivation of the dyna mic equations for the scale factor R(t) in the Friedmann -Robertson-Walker metric of the uni verse in General Relativity. His starting point is a simple argument leading t o       󰇛      󰇜  V is the Hubble vol ume,     , where H is the Hubble cons tant, a nd L p is the Planck lengt h. Th e right-hand side is ze ro for the de Si tter universe, which i s predicted to be the asymptot ic state of our current universe. He presents an a rgument that the cosmological constant is related to the Planck le ng th , lea ding to the expression where  i s of order un ity and gives a val u e consiste n t with observation. 13. Complexity theory Comple xity and emergence are often used interchan geabl y , as discussed earlier. S ince t h e 1980’s a fiel d known as complexity theory or complexity science has develope d 244,245 with associat ed conferences, journal s, and instit u tions. Je nsen has written a n aca demic monograph that sta tes what de f ines the fiel d is the emphasis on networks. 6 Popular books have be en published by Gleick, 82 Holland, 22 Mitchell, 246 Parisi, 247 Wal drop, 248 and Coveney and Highfiel d. 249 A few topics that I have already discussed include chaos th eory, neural networks, and pat tern formation. Below I briefly discus s s ome othe r relevant topics. Some will oc cur again in discussions of biology and economics. Networks These a re also known as graphs : large collections of verti ces connected by edges. D ifferent forms of networks that ha v e been studied extensive ly include sc ale-free and small-world networks and the y have been rev iewed by Albert and Barabasi, 250 Newman 80,251,252 and Strogatz. 253 These networks have attracted considerable interest because t hey o ccur in many biologi cal and social systems. Scale-free net works h ave the property that P(k), the probability of a node bei ng connec ted to k other nodes scales like a n inverse power of k. This means there i s a high probability of the network having hub s, i.e., particul ar nodes that are connec ted to a large number of other nodes. S uch hubs represent modularity at the me soscale. Stati stical methods have been developed to identify s uch hubs and their associated comm unities in r eal n etworks. 254 This ma ny reveal o verlappi ng comm un ity structure. 254 Power laws Exam ples include fractals, Zipf’s law and the Pareto distribut ion. These have been reviewed by Newman 80 and are the subje ct of a popular book by West. 255 Cell ular automata Conway's G ame of L ife is a popular and widely studied version of cellular aut omata. It is based on four simpl e rules for the evolution of a two -dimensional grid of squares that can eit her be “ dead ” or “ a live . ” Distinct complex patterns can emerge, including still lifes, oscill ators, and spaceships. Blund ell argued that the Game of Li f e 256 illustrate d emergence, considering t h e "scatte r ing" of different objects off o ne anot her could lead to th e creation and destruct ion of objects s uch as space ships, "Canada geese", and "pulsars". Wolfram p erformed a n exhausti v e study of one-dime nsional a utomata where bits are updated based only on t he st ate of their two nearest neighbou rs. H e found 257 that the 256 different autom ata of this type, and their be h aviour, falls into four universality classes: uniformity, periodi c time dependence, fractal, and co mplex non-repetitive patterns. Wolfram spe culated that t h e latte r may perform un iversal computati on. “Order at the edge of chaos.” This concept has been heavily promoted 248 bu t is contenti ous. In the c ontext of evolution, Kauffman conjectured that when a biological system must perform com pu tations to survive, natural selection will favour systems near a phase transition betwee n chaos and order. The concept has also been invoked to understand an d improve inte rna tional ai d programs. 258 Packard conside r ed how the rul es of cellular automat a (CA ) might e vo lve according t o a genet ic algorithm whi ch is characte r ised by a pa r ameter, lambda, de fined by Langdon. According t o Mitchell et al. 259 Packard made “ two h ypotheses: (1) CA rule s able to perform compl ex computations are most likely to be found near “ critical ” lambda values, which have been c laimed to correlate with a phase transition betw een ordere d and chaotic behavioural regime s for CA; (2) Wh en CA rules are evolved to perform a complex computation, evolut ion will tend to select rules with lambda values close to the critical val ues. ” These have been c ontested by Mit chell et al. 259 Crutchfield pointed out 260 that the observation that in several specific models that the intri nsi c computational capacity is maximised at phase transit ion led to the conjecture that the r e is a universal interdepe nd ence of ra ndo mness and structure. T h is led t o the hope that there was a single universal complexity-entropy function. However, this turne d out not to be the case. Self-reproduci ng machines Until t he 1950’s it was ass umed t hat a machine could not reproduce itself and that this was the fundament al difference between machines and living systems. However, von Neumann 261 showed that t his was incorrect. In the process, he invented t h e concept of a cell ular autom aton. The on e he studied was two-dimensional and each cell could i n on e of 29 different st ate s. S elf-replication is an emergent prope rty. The pa r ts of the machi ne do not have t his property. The whol e mac h ine onl y h as this property if it has enough parts and the list of i nstructions is long enough. Quan titative cha n ge produces qualitative change. Brenner argued 262 that von Neumann’s work, toget h er with t h at of Turi ng on un iversal comput ation, cent ra l to bi o logical theory. Self-organized c ri ticality This is anot her h eavily promoted conce p t that has been claime d to be an orga n ising principle for comple x systems 93 but has remaine d contentious. Bouchaud reviewed 263 the releva n ce of the idea of se lf-organised critical ity for finan ce and econom ics: " The semi n al i d ea of Per Bak is to think of model parameters themselves as dynami cal variables, in such a way that the system spontaneously e vo lves towards the critica l point, or at lea st v isits it s ne ighbourhood frequently enough ." A key property of systems exhibi ting criticality is power laws in the probability distribution of a property. Next, he reviewed the critical branching transition, a toy model that describes diverse systems incl uding “ s and pile avalanc hes, brain activity, epide mic propa g ation, default/bankruptcy waves, word of mouth, ..." This ill ustr ates unive rsa lity. The model involves the parameter R 0 which bec ame famous during the COVID -19 pandemic. R 0 is the average number of uninfected pe ople who become infected due to conta ct wit h an infected individual. For sand pile s R 0 is the average number of grains that st art rolling i n response to a single rolling grain. When R 0 < 1, a single unstable grain will on average only dislodge 󰇛     󰇜  grains. Very large a v ala nches have an exponentially small probability. As R 0 approaches 1, the probability P(S) of an avala nche involving a large number of gra ins S is given by 264  󰇛  󰇜      󰇛  󰇛     󰇜   󰇜 When R 0 = 1 the distribution of avalanche size s is a s cale-fr ee, power-la w distr ibution 1/S 3/2 , with infi nite mean. Bouch aud observed that "most avalanches are of sma ll size, although some ca n be very large. In other words, the system looks stable, but occasionally goes haywire wit h no apparent cause." 14. Chemistry It is im portant to be clear what the system is. Mos t of chemistry is not really about isolated mole cules. M ost chemical reactions happen in an environment , of ten a solvent. Th en the system is the c h emical s of in terest and the solvent. For example, when it is stated that HCl is an ac id, this is not a r eference to i sol ated HCl molecules but a solution of HCl in water which dissociat es into H + and Cl - ions. Chemical properties s uch as reac tivity can change signific antly depending on whether a compound is in the soli d, liquid, or gas state, or on a solid surface or dissolved in a solve n t. Scale s Rele vant scales include the total numbers of atoms in a compound, which can range from two to mi llions, the total number of electrons, and the number of different chemi cal elements in the c ompound. As the number atoms and electrons increases so does the di mensionalit y of the Hil bert space of the corresponding quantum system. The t ime scales for processes, which range from molecula r v ibrations to chemical reacti ons, can va ry from femtoseconds to days. Relevant e n ergy s cale s, corresponding to differe n t effective interactions, can v ary from tens of eV (strong covale nt bonds) to microwave energi es of 0.1 m eV (molec u lar rotational energy level transitions ). Novel ty All che mical compounds are composed of a d iscrete number of atoms, usually of differen t type. F or exam p le, acetic acid, denoted CH 3 COOH , is composed of c arbon, oxygen, and hydrogen atom s. A compound usually has chemical and physical properties that its constit uent ato ms do not have . Ch emistry is all about transforma tion. Reactants combine to produce product s, e.g., A + B -> C. Th e product C may have chemical or physical properties that t h e reacta nts A and B did no t have. Chemi stry involves concepts that do not appear in physics. Hoffmann argued 265 that concepts such as acidity and ba sicity, aromaticity, functional groups, and substituent e ff ects have gr eat util ity and ar e lost i n a reductionist pe rspective that tries to define them precisely and mat hematicise them. Diversity Chemi stry is a wonderland of div ersity as it puts chemical eleme n ts in a multitude of different arrangement s tha t produce a plethora of phenomena. For example, much of organic chem istry involves only three different atoms: carbon, oxygen, and hydrogen. Molec ular structure Simple mo lec u les (such as water, ammonia, carbon dioxide , methane , benzene) have a unique structure de f ined by fi xed bond lengths and angles. I n other words, there is a well-defined geome tric structure that gives the locations of the centre of atomic nuclei in the molecule. This structure is a classical entity that emerges from the interactions betwee n the electrons and nucl ei of the constituent atoms , and the d ecohering effect of the molecular environment . In philosophic al discussions of emergence in chemistry, mol ecular structure has received signific ant attention. 11,266 Some claim it provides evidence of st rong e merge n ce (to be discussed lat er in the section on philosophy). The arguments c entre around the fact that the mole cular structure is a classical entity and conce p t that is imposed, whereas a logically self- consistent approach would treat both electrons and nuclei quantum mechanically, a llowing for their qua n tum e n tanglement. 267,268 The m olecular structure of ammon ia (NH 3 ) ill ustr ates the issue. It has an um br ella structure which ca n be inverted. Classically, t h ere are two possible de generate structures. For an isolat ed molecule qu antum tunnelling bac k and forth between the two structures can occur. The ground sta te is a quantum superposition of two molec u lar structures. This tunnelling does occur i n a d ilute gas of ammonia at low temperature, and the a ssoci ate d qu antum transition i s the ba sis of the maser, the forerunner of the laser. At higher temperatures and i n denser gases the t unneling is washed out by d ecohere n ce and the nuclear wavefuncti on collapses onto a defini te s tructure. This example of ammonia was dis cussed by Anderson at the be g inning of his semina l More is Diffe ren t article 25 to illustrate how s ymmet ry breaking leads to well- define d molecular structures in large molecules. In different words, the problem of molec u lar structure i s intricately connected with that of the quantum-class ical bound ary. Born-Oppenheimer approximati on Wit hout this concept, much of theoretical chemistry and condensed matter would be incredi b ly difficult . I t is based on the se p aration of time and e n ergy scal es associated with ele ctronic and nuclear motion. p It is used to de scribe and understand the dynamics of nucl ei and el ectronic transitions in solids and molecules. The potential energy surface s for diff erent ele ctronic states define effective theory for the nuclei. Without this concept, much of theoretic a l chemistry and c ond ensed matte r would b e incre d ibly di ff icult. Singularit y. The Born-Oppenheimer a pprox imation is justifie d by an asymptotic expansion in powers of (m/M) 1/4 , where m i s the mass of an electron and M the mass of an atomic nucle us p The Born-Oppenhei mer approximation is an example of a general approach to quantum mec hanics problems, discuss ed by Migdal . 269 Consider a system composed of two subsystems that ha ve dynamics on two vastly different time scales, termed fast and slow. T he effect s of th e fast system on the slow system can be treating by adding a potential e n ergy term to the Hamiltonian operator of the slow system . in the molecule. Th e significance of this singularity for understanding emergence has been discussed by Primas 54,270 and Bishop. 11 The rotational and vi brational degre es of freedom of molec u les also i nvo lve a sepa r ation of tim e and energy scales. Consequently, one can derive separate effective Hamiltonians for th e vibrati on al a nd ro tational de gr ees of freedom. Qualitat ive difference with increase in molecular size Consider the following serie s with v arying chemical propertie s: for mic acid (CH 2 O 2 ), acetic aci d (C 2 H 4 O 2 ), propionic acid (C 3 H 6 O 2 ), butyric acid (C 4 H 8 O 2 ), and vale rianic acid (C 5 H 10 O 2 ), whose membe rs involve the successive addition of a CH 2 radical. The Marxist Fri edrich Engels used these examples as evidence for Hegel’s law: “ The law of transforma tion of quantity into quality and v ice versa ” , that was discussed earlier. In 1961 Platt 271 discussed properties of large molecules that “might no t have been anti cipated” from properties of their chemical subgro ups. Table 1 in Platt’s paper listed “Properti es of mo lecules in the 5- to 50- atom range that have no counterpart in diatomics and many t riatomics.” T able 2 listed “ Proper ties of molecules in the 50- to 500- atom range and up that go be yond the propertie s of their chemical su b -groups .” The properties li st ed incl ud ed inte rna l conve rsion ( i.e., non-radiative decay of excited electronic stat es) , formation of mic elles for hydrocarbon chains with more than ten carbons, the helix-coil tra nsi tion in polyme rs, chrom atographic or molec ular sorting pro pertie s of poly electrol y tes such as those in ion-exc h ange resins, and the cont r actility of long chains. Platt also discuss ed the problem of mo lecular self-replication. As discussed earlier, until t he 1950’s it was assumed tha t a machine c ou ld not reproduce it se lf and this was the fundamental difference bet ween machines and living systems. Ho w ever, von Neuma nn showed that a mac hine with enough p arts and a sufficiently long list of i nstructions can reproduce itself. Platt pointed out that this suggest ed there is a thresho ld for autocatalysis : “this threshold marks an e ssentially discontinuous change in properties, and that molecules larger than this size di ffer fro m all smal ler ones in a property of cent ral im portance for biology.” Thus, s elf- replicat ion is an emergent property. A modification of this idea has been pursued by Kauff man with regards to the origin of life, 3 that when a network of chemical reactions is sufficie ntly large it b ecomes self-replicating. Coarse graining in multi-scale modelling Computa tional chemists now routine ly study system s containi ng as many as billions of atom s. D ynamical si mulations on such large systems are restricted to treati ng the atoms cla ssically mov ing in effect ive potentials that known as “force fields”. These are all paramete rised and so me are d ete r mined by electronic structure calculations using the Born- Oppenheim er approximation. Gi ven the large number of atoms, the goals of co mputational efficiency and physical i nsight have l ed to the development of a range of coarse -graining met hods. Th ese have recently bee n reviewe d by Jin e t al. 67 Basically, sub-groups of atoms are r eplaced by “ b lobs” that interact with one a nother via a new effective potential. Consider a system, de f ined by Fine-Grained (FG) a set of n coordinates r n , and interacting with one a nother via a potential u FG ( r n ). Coarse-Graining involves defining a set of N β c , i.e. [the average e qu ilibrium value of S i ] ϕ  ≠ 0 even in the a bsence of any indivi dually preferred choice (i.e., F=0). When F≠0, one of the t wo equ ilibria is e xponentially more probable tha n the ot her, and in principle the population should be locked into the most like ly one: ϕ  >0 whenever F>0 and ϕ  <0 whenever F<0. Unfortunate ly, the equilibrium analysis is not sufficient to dra w such an op timistic concl usion. A more detailed analysis of the dynamics reveal s that the time needed to reach equil ibrium is exponentially large in the number of agents . As noted by Keynes, "in the long run, we are all de ad." This situation is well-known to physicists but is not so well appreciated in othe r c ircles. For example , it is not discussed by Brock and Durlauf. Bouchaud discuss ed the m eta-stab ility associated wit h the two possible polarisations, as occurs in a first -order phase tra nsition. From a non-equ ilibrium dynamical analysis, based on a Langevin equation, the t ime τ n eeded for the system, starting around ϕ =0, to reach ϕ  ≈1 is given by:   exp[  (1−  /  )], where A is a numerical factor. This means t hat whenever 0