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Reaction-Diffusion Processes, Critical Dynamics

arXiv:hep-th/9302112v1 23 Feb 1993Reaction-Diffusion Processes, Critical Dynamicsand Quantum ChainsFrancisco C. Alcaraza1, Michel Drozb, Malte Henkelb and Vladimir Rittenbergaa Physikalisches Institut, Universit¨at BonnNußallee 12, D - 5300 Bonn 1, GermanybD´epartement de Physique Th´eorique, Universit´e de Gen`eve24 quai Ernest Ansermet, CH - 1211 Gen`eve 4, SwitzerlandUGVA-DPT 1992/12-799AbstractThe master equation describing non-equilibrium one-dimensional problemslike diffusion limited reactions or critical dynamics of classical spin systemscan be written as a Schr¨odinger equation in which the wave function is theprobability distribution and the Hamiltonian is that of a quantum chain withnearest neighbor interactions. Since many one-dimensional quantum chains areintegrable, this opens a new field of applications.

At the same time physicalintuition and probabilistic methods bring new insight into the understanding ofthe properties of quantum chains. A simple example is the asymmetric diffusionof several species of particles which leads naturally to Hecke algebras and q-deformed quantum groups.

Many other examples are given. Several relevanttechnical aspects like critical exponents, correlation functions and finite-sizescaling are also discussed in detail.1Permanent adress: Departamento de F´ısica, Universidade Federal de S˜ao Carlos, 13560 S˜aoCarlos SP, Brasil

1IntroductionOur understanding of nonequilibrium statistical physics is far behind that for theequilibrium theory. Even simple models may pose a formidable problem if one wantsto approach them analytically.

In this paper, we shall consider two different types ofsuch problems: the diffusion-limited chemical reactions and the critical dynamics ofclassical spin systems. It will be shown that in one dimension these problems can bemapped onto quantum chain problems which are often integrable and on which a lotof progress was recently achieved [1].

As a result, many new predictions concerningthe nonequilibrium statistical physics problems follow.The study of the diffusion-limited chemical reactions has stimulated a vast amountof research since the first investigation of Smoluchowski many years ago [2]. Examplesare given by the bimolecular reactions, A + Bk⇀↽g C + D where the two species A andB diffuse and react to form the two new species C and D and k and g are the forwardand backward reaction rates, respectively.

The simple case of irreversible reactionsfor which g = 0, C is a inert product and D is not present has been extensivelyinvestigated since the original work of Zeldowich [3]. Despite their simplicity thosesystems exhibit a very rich dynamical behaviour.

For homogeneous initial conditionsand in low dimensions, the diffusion mechanism is not efficient enough to mix theparticles. As a result a spatial segregation occurs and accordingly, a reduced numberof reactions between the two species is possible.

This results in a slowing down ofthe dynamical evolution called anomalous kinetics [4]. The evolution of the systemis not properly described by the usual rate equations, since the fluctuations play acrucial role.

Another interesting situation is when the reactants are initially separatedin space; then a reaction-diffusion front is formed during the evolution [5].Hereagain the properties of the front are drastically influenced by the fluctuations in lowdimensions [6]. We shall restrict ourselves to the homegeneous case here.

Severalapproaches have been used to study such systems: numerical simulations [7], scalingarguments [6] and analytic arguments based on the theory of the Brownian motion [8].The main results obtained concern the decay of the number of particles of one species[9] and the temporal evolution of the gap developing between the particles of the twospecies due to the segregation process [10]. However, the analytic results are scarce.Very little is known concerning the behavior of quantities like the space and/or timedependent two-particle correlation functions.

Accordingly, new theoretical approchesallowing the computation of such quantities are desirable.Let us first introduce several models which have been studied in the literature:1. The coagulation model: [11] one considers molecules of one species (say A), thatdiffuse in a milieu and react as:A + A →A(1.1)2.

The annihilation model: [12] the A molecules diffuse and annihilateA + A →∅(1.2)1

where ∅denotes an inert state which decouples completely from the dynamics.3. The two-species trapping reaction: [13] two types of molecules A and B diffuseand one of them, B, is “trapped” by A:A + B →A(1.3)4.

The two-species annihilation reaction: [9] two types of molecules A and B diffuseand annihilateA + B →∅(1.4)We are interested in the long-time behaviour of systems like the four examplesgiven above. Considering quantities like the mean concentration of A particles cA(t),in general we expect the following two types of behaviour, as t →∞cA(t) ∼(t−αexp −t/τ(1.5)where α is some constant and τ is known as relaxation time.

Throughout this paper,we shall refer to the first type as “massless” or “critical”, while the second case will bedenoted as “massive”. This terminology is borrowed from field theory and equilibriumstatistical mechanics.

All four reactions defined above have a critical behaviour inthe sense that for a given initial concentration of particles, the long time behaviourof the concentrations has an algebraic fall-off. For example, in the reactions (1.1) and(1.2) and in one-dimension, the concentration cA(t) of particles behaves likecA(t) ∼t−1/2(1.6)One can allow for reversible (or back) reactions, corresponding to g ̸= 0.

In thelong time limit (t ≫1/g), a local equilibrium state is reached [14]. One is then in a“massive” regime, in which the relaxation towards equilibrium is exponential.A different type of nonequilibrium problem is the one of critical dynamics ofclassical spin systems.

Let us consider for example the case of a classical Ising modelon a lattice. The system is prepared in an initial (nonequilibrium) state.

One wouldlike to know how fast the equilibrium state (Gibbs state) will be reached. When not at(static) criticality, the system relaxes exponentially towards equilibrium.

In general,the relaxation time τ scales with the spatial correlation length ξ as the temperatureapproaches its critical valueτ ∼ξz(1.7)where z is the dynamical critical exponent. However, counterexamples will be given inthis paper where ξ diverges but τ remains finite (no critical slowing-down!) As theseclassical spin systems do not have an intrinsic dynamics, the dynamics is thoughtto come from the interactions between the spins and a heat bath modelling the fast2

degrees of freedom not included in the classical Hamiltonian. The dynamics of thesemodels is thus formulated in terms of a master equation for the probability that aspin configuration is realized at time t [15].

Several cases have to be distinguisheddepending on the presence of macroscopic conserved local quantities. The simplestcase is the purely relaxational one in which there are no local conservation laws and weshall restrict to this case throughout the paper.

For example, the initial magnetizationrelaxes towards its equilibrium value. All the physics is put into the transition ratesappearing in the master equation.

Several choices are possible compatible with thecondition of detailed balance, which is the condition insuring that the stationary statewill be the Gibbs one. Despite its apparent simplicity, the solution of this problemis very difficult and little can be said analytically even in one dimension [16].

Themapping onto a quantum chain Hamiltonian, which will be explored throughout thepaper, will turn out to be a very useful tool to clarify some controversies present inthe literature [17], [18].Both types of nonequilibrium problems (diffusion-limited chemical reactions andcritical dynamics) can be described in terms of a master equation for P({β}, t), theprobability that a configuration {β} of the systems is realised at time t. It turnsout that it is suitable to map this master equation problem onto a quantum chainproblem [16, 19, 20]. The corresponding equation of motion reads∂tP({β}, t) = −HP(1.8)where H is directly related to the transition rates appearing in the master equation.This will be detailed in Section 3.

The question of knowing what kind of dynamicalbehaviour has the model (power law or exponential relaxation) amounts to know inwhich phase (massive or massless) of the phase diagram we are.Moreover, in reaction-diffusion processes H turns out to be non-hermitian andhas often the following particular structureH = H0 + H1(1.9)where H0 is a known integrable Hamiltonian with a larger symmetry than H (forexample, in two-states models it can be the XXZ quantum chain in a Z field). H1is non-hermitian and has a lower symmetry but does not affect the spectrum of H.Thus, if H0 is massless, it follows that H is massless as well.

In problems of criticaldynamics, H has again often the structure of Eq. (1.9) but with a new meaning.

His now hermitian, H0 is again a known Hamiltonian (it can be the same as the oneoccuring in reaction-diffusion processes) but H1 is now a perturbation term, if weapproach criticality (small temperatures).When writing this paper we were faced with two problems. The first one wasthat we realized that we are left with many more open questions than answers.

Thisis kind of nice because we hope that this is an invitation for other people to lookcloser at the subject. The second problem concerns the pedagogical presentation ofthe paper since it addresses two different communities: people doing nonequilibrium3

statistical mechanics and who are familiar with the physical problems treated inthis paper but not with integrable systems and those familiar with the Bethe ansatzand two-dimensional field theory but not familiar neither with the physical problemsdiscussed here nor with methods of computing nonequilibrium averages which aredifferent from those techniques used in equilibrium statistical mechanics (vacuumexpectation values). We thus suggest two approaches to this paper.

One for the“mathematician”, the other for the “physicist”.The “mathematician” should start with Appendix C (the last chapter of the pa-per). There we remind the reader of the definition of the Hecke algebra which dependson a parameter β = q +q−1 (the significance of q will become apparent immediately).As is well known, if a quantum chain can be written as a sum of generators of a Heckealgebra, through Baxterization [21] one can associate to the chain an integrable ver-tex model.

In this paper, we will consider only chains with 2L and 3L states ( L is thelength of the quantum chain). Accordingly, we are going to look for some quotients ofthe Hecke algebra.

To various quotients of the Hecke algebra one can associate a rep-resentation given by the (m/n) Perk-Schultz quantum chain [22]. These are chainswith (m + n)L states invariant under the quantum superalgebra UqSU(m/n) [23].In this paper, the (2/0), (1/1), (3/0) and (2/1) Perk-Schultz models will play a rolewith q real (|β| ≥2) and the physical significance of the deformation parameter q willbecome apparent.

We also give some new representations of the quotients. First wegive non-hermitian representations (for q real), these are relevant for expressions likeEq.

(1.9) and next, we give representations of the (2/0) and (1/1) quotients with 3Lstates. Notice that in the last case the symmetries of the corresponding chain are notanymore UqSU(2/0) and UqSU(1/1).

The knowledge of these symmetries is impor-tant because if two chains belong to the same quotient, their spectrum is in generalthe same but the degeneracies are fixed by the symmetries (a more detailed versionof Appendix C is going to be published elsewhere). After finishing this appendix, the“mathematician” should go through Sections 2-4 and have a close look at Section 5where, in the simple case where the calculations can be done using free fermions,one illustrates the peculiar problematics of nonequilibrium statistical mechanics.

Itis stressed that (see Eq. (1.8)) the knowledge of the wave function (not only the spec-trum) plays a crucial role and since in the Bethe ansatz this knowledge is hard to get,the calculation of average quantities presents a new challenge.

The “mathematician”should next have a look a Appendices A and B and skip Sections 7-9.We sugggest to the “physicist” to read the paper in the chronological order. InSection 2 we consider quantum chains with L sites (we always take open chains).On each site we take a discrete variable β taking N values (β = 0 corresponds to avacancy).

We write the most general master equation describing bimolecular reac-tions. In Section 3 we write down the corresponding one-dimensional Hamiltonian,see Eq.

(1.8). A close related development connecting the master equation in discretetime to the transfer matrix formalism can be found in Refs.

[24].Two-states models are considered in Section 4. The elementary processes in themaster equation describe besides diffusion, annihilation, coagulation and death pro-4

cesses (these processes lead for large times to a state with vacancies only) also thereverse processes creation, decoagulation and birth.We first show that for purediffusion processes which are left-right asymmetric, the Hamiltonian is just the q-deformed XXZ spin-1/2 Heisenberg chain.One discusses in detail the quantumchains corresponding to the different processes and one stresses the importance ofthe phase diagram of the XXZ Heisenberg chain in a Z magnetic field, especially thePokrovsky-Talapov line. The learned reader will also notice the importance of non-hermitian representations of the braid group occuring in this type of problems.

Asis well known when a forward-backward process exists (annihilation and creation forexample) through a similarity transformation, the Hamiltonian is hermitian. Whenall three forward-backward processes are allowed, this is not always possible.

Wederive the conditions on the rates in oder to get hermitian Hamiltonians.Section 5 illustrates in detail the simple example of annihilation only, with a rateequal to half the diffusion constant. This corresponds to the physical picture in which,when two particles are on neighboring sites, they always annihilate.

As was alreadyknown, in this case all the calculations can be done using free fermions. Here webring some new results.

We consider the finite-size scaling of the problem (large timeand lattice size with z = tL−2 fixed). We show that the finite-size scaling functionexists (no logarithmic corrections).

This result is important because if one accepts itsgeneral validity, it allows numerical estimates of critical exponents like in equilibriumproblems. We also compute, for the first time, using the Hamiltonian formalism thedensity-density correlation function and stress the importance of the scaling limitr, t →∞with u = r2/t fixed.In Section 6 we consider three-state models (two species of particles and vacancies)with Z3 symmetry.

Various integrable quantum chains occur which allows us to obtainsome rigrous results. Nevertheless, as will be seen much work has still to be done.An interesting three-states model with Z2 symmetry is mentioned in Appendix C.In Section 7 we derive and solve the condition on the rates in the master equationwhich gives chosen steady states.

Based on these results, in Sections 8 we discuss thedynamics of the Ising and chiral Potts models. This takes us back to the quantumchains discussed in Sections 4 and 6.We look at the behaviour of the systemswhen the temperature is small (critical dynamics).

This brings us to a problem ofperturbation theory. Two cases occur.

In the first, although the (spatial) correlationlength diverges, the relaxation time stays finite. If we take the corresponding rateto zero and get critical dynamics, then the relaxation time depends on the (spatial)correlation length in a way which is independent of the remaining rates (universality).In Appendix A we study closer Eq.

(1.9). We show how from the knowledge ofthe eigenvalues and eigenvectors of H0 one can compute the eigenvectors of H. InAppendix B we consider the example studied in Section 5 from a different point ofview.

We notice that the non-hermitian Hermiltonian corresponds to a representationof the Hecke algebra and using Baxterization, we derive the corresponding vertexmodel. It turns out that this is a seven-vertex model.

This observation is relevant,since as shown in Appendix C (see also [25]) there are other examples of non-hermitian5

chains (irreversible processes! )which satisfy the Hecke algebra and through thisprocedure one can find the wave functions using the Bethe ansatz.

Section 10 closesthe paper with some open questions.2The master equationIn order to write the master equation which describes a general lattice version ofa reaction-diffusion process in one dimension, we take a chain with L sites and ateach site i we take a variable βi taking N integer values (βi = 0, 1, . .

. , N −1).

Byconvention we attach the value βi = 0 to a vacancy (inert state). We want to considera master equation for the probability distribution P({β}; t) with the following form∂P({β}; t)∂t=L−1Xk=1"−w(k)0,0(βk, βk+1)P(β1, .

. .

, βL; t)+N−1X′ℓ,m=0w(k)ℓ,m(βk, βk+1)P(β1, . .

. , βk + ℓ, βk+1 + m, .

. .

, βL; t)(2.1)where w(k)ℓ,m are the transition rates and the prime in the second sum indicates thatthe pair ℓ= m = 0 should be excluded. We assume all the additions performed onthe βi to be done modulo N.The advantage of this notation is that one can introduce discrete symmetries in asimple way.

We shall assume hereafter that the system is homogenous which impliesthat the transitions rates are independent of kw(k)ℓ,m(α, β) = wℓ,m(α, β)(2.2)for all k = 1, . .

. , L −1.

The probability Γ(γ,δ)(α,β) that a state (γ, δ) on two consecutivesites will change after an unit time into the state (α, β) isΓ(γ,δ)(α,β) = wγ−α,δ−β(α, β) ; (α, β) ̸= (γ, δ)(2.3)The rates w0,0(α, β) are related to the probability that in the unit time the state(α, β) unchanges. From the conservation of probabilities, we havew0,0(α, β) =Xr,s′wr,s(α −r, β −s)(2.4)where r = s = 0 is again excluded.It is now trivial to check, using Eq.

(2.3), that if for the N-state model we wantto have a ZN symmetry, only the functionswℓ,m(α, β) , ℓ+ m = 0(modulo N)(2.5)will appear. This is the case of the annihilation model ( written as A + A →∅+ ∅where ∅is a vacancy) which has Z2 symmetry (N = 2) and for the two-species6

annihilation reaction (N = 3 and the reaction is written as A+B →∅+∅) which hasZ3 symmetry. In the former case, we assign to the vacancy the Z2 quantum number0 and to the A molecule the number 1.

In the latter case, the vacancy is denoted by0, the A by 1 and the B by 2.Parity conservation (left-right symmetry) is achieved ifwℓ,m(α, β) = wm,ℓ(β, α)(2.6)Let us comment on the supplementary symmetries besides a possible parity invari-ance which exists in the four examples given in Section 1. The coagulation model hasno symmetry at all.

As we have seen, the annihilation model has a Z2 symmetry. Thetwo-species trapping reaction has a U(1) symmetry (since the number of A particlesis conserved) and a Z2 symmetry (give to the vacany and the state B a Z2-parity “+”and to the state A a Z2-parity “-”).

As mentioned before, the two-particle reactionhas a Z3 symmetry. It has also a supplementary U(1) symmetry, since the differenceof the numbers of A and B particles is conserved.It is often useful to make a change of variables in the master equation.

In thispaper, we defineP({β}; t) = Φ({β})Ψ({β}; t)(2.7)where Φ({β}) takes the special formΦ({β}) =LYk=1h(k)(βk)(2.8)From Eq. (2.1) Ψ is a solution of the new master equation∂∂tΨ({β}; t)=L−1Xk=1"−w0,0(βk, βk+1)Ψ({β}; t)+Xℓ,m′W (k)ℓ,m(βk, βk+1)Ψ(β1, .

. .

, βk + ℓ, βk+1 + m, . .

. , βL; t)(2.9)with the pair ℓ= m = 0 excluded andW (k)ℓ,m(α, β) = ϕ(k)(α + ℓ, β + m)ϕ(k)(α, β)wℓ,m(α, β)(2.10)whereϕ(k)(α, β) = h(k)(α)h(k+1)(β) ; k = 1, 2, ..., L −1.

(2.11)Notice that although the rates wℓ,m(α, β) are link independent in general, the W (k)ℓ,m(α, β)are not. The function Φ({β}) has obviously to be non-zero and finite.

Let us finallynote that if we want all “molecules” to disappear at large times so that we are onlyleft with vacancies, we must have the conditionwℓ,m(−ℓ, −m) = 0(2.12)7

for ℓ, m = 0, 1, . .

., N −1. One can check thatP({β}; t) =L−1Yk=1δ(βk)(2.13)satisfies indeed ∂P/∂t = 0, with the convention that βk = 0 corresponds to a vacancy.3Quantum chains corresponding to master equa-tionsWe shall now write the master equation (2.9) in the form of a Schr¨odinger equation.In order to do so, on each site we define a basis in the space of N × N matrices Ekℓ.The only non-vanishing matrix element of the matrix Ekℓis the one on the kth lineand the ℓth column and this matrix element is equal to unity.

Assuming homogeneity,the most general NL × NL Hamiltonian with only nearest-neighbor interactions canbe written asH =L−1Xi=1Hi(3.1)whereHi =Xaℓ,m,r,sEℓm ⊗Ers(3.2)acts in the subspace V (i) ⊗V (i+1) of the NL-dimensional vector space. Besides thematrices Ekℓ, on each site it is also convenient to define the matrix FF =0100· · ·00010· · ·0..................000· · ·01100· · ·00; F N = 1(3.3)With these notations, the Hamiltonian density can be written asHi = Ui −Ti(3.4)whereTi=N−1X′ℓ,m=0N−1Xα,β=0W (i)ℓ,m(α, β)EααF ℓ⊗EββF m(3.5)Ui=N−1Xα,β=0w0,0(α, β)Eαα ⊗Eββ(3.6)8

Here W (i)ℓ,m is given in Eq. (2.10) and w0,0 in Eq.

(2.4). The Schr¨odinger equationreplacing the master equation reads∂∂t |Ψ⟩= −H |Ψ⟩(3.7)The Schr¨odinger equation corresponding to the physical problem (without the simi-larity transformation Eq.

(2.7)) is∂∂t |P⟩= −fH |P⟩(3.8)fH is obtained by taking wℓ,m(α, β) instead of Wℓ,m(α, β) in Eq. (3.5).

Obviously Hand fH have the same spectra but they are given by two different one-dimensionalquantum chains which act in a NL Fock space (for N −1 species) and their explicitform depends on the chemical reaction and diffusion process. We first discuss theSchr¨odinger equation (3.8) The ket state |P⟩is defined as follows [20].Take anorthogonal basis in {β}|{β}⟩= |β1, .

. .

, βL⟩; ⟨{β′}| |{β}⟩= δ{β′},{β}(3.9)then|P⟩=X{β}P({β}; t) |{β}⟩(3.10)The reaction-diffusion process is determined by the initial (t = 0) probability distri-bution P0({β}) which defines the “initial” ket state|P0⟩=X{β}P0({β}) |{β}⟩(3.11)The Hamiltonian fH is in general non-hermitian and due to probability conservation(Eq. (2.4)), it satisfies the remarkable relation:⟨0| fH = 0(3.12)where the bra ground state ⟨0| is⟨0| =X{β}⟨{β}|(3.13)From Eq.

(3.12) it follows that the ground state energy is zero. Take now an observ-able X({β}) (for example the concentration of A particles in the coagulation model(1.1)).

Its average can be computed as follows< X > (t)=X{β}X({β})P({β}; t)=⟨0| X |P⟩= ⟨0| Xe−eHt |P0⟩(3.14)9

Notice that that in nonequilibrium problems one studies the properties of the wavefunction which is already a probability and not quantum mechanical expectationvalues⟨0| X |0⟩(3.15)as one does in equilibrium problems. The thermodynamical (continuum) limit canbe computed from Eq.

(3.14), taking the length L of the chain to infinity for a fixedtime t. As discussed in detail in Section 5, a second limit (the finite-size scaling limit)is also interesting, where one takes both t and L large but keeps z = t/L2 finite.If Eλ and |Ψλ⟩are the eigenvalues and eigenkets of fH, we have from Eq. (3.14)< X > (t) =Xλaλe−Eλt ⟨0| X |Ψλ⟩(3.16)where|P0⟩=Xλaλ |Ψλ⟩(3.17)Thus the large t behaviour of < X > (t) is governed by the lowest excitations of fH.If, instead of fH we use H (see Eq.

(3.7)), then the averages have different expressions< X > (t) =X{β}X({β})Φ({β})Ψ({β}; t) = ⟨0| XΦe−Ht |Ψ0⟩(3.18)where|Ψ0⟩=X{β}Φ−1({β})P0({β}) |{β}⟩(3.19)There is another distinctive feature of nonequilibrium problems, as compared toequilibrium ones, and this is the concept of interaction range. As opposed to equilib-rium problems where the whole information is only contained in the Hamiltonian (itis nearest-neighbour interactions as can be seen from Eq.

(3.1)), in the nonequilib-rium case we have to give also P0({β}). This probability can describe an uncorrelatedhomogenuous distribution likeP0({β}) =LYi=1f(βi)(3.20)or a strongly correlated distribution when, for example, at t = 0 the reactants areseparated in space [6].

The general properties of P({β}; t) and implicitly those ofaverages (like self-organization, critical dimensions or critical exponents) are goingto be different. This can be understood when comparing Eqs.

(3.14) and (3.18). Letus assume that we give fH and P0 describing a correlated distribution, we can makea similarity transformation to bring P0 to an uncorrelated distribution Ψ0.

Afterthis transformation we will have to work with a new Hamiltonian H (in general withlong-range interactions) with different physical properties.10

4Two-state HamiltoniansSince for two-states models instead of the basis Eα,β one often prefers the basis ofPauli matrices, let us start by giving some useful identitiesE01 ⊗E10 + E10 ⊗E01=12 (σx ⊗σx + σy ⊗σy)E01 ⊗E01=14 (σx ⊗σx −σy ⊗σy + i (σx ⊗σy + σy ⊗σx))E01 ⊗E01 + E10 ⊗E10=12 (σx ⊗σx −σy ⊗σy)(4.1)We are now going to consider various Hamiltonians according to their symmetriesand chemical properties. In the two-states models, we have A states and vacancies.4.1Z2 symmetric, parity non-invariant vacuum-driven pro-cessesBy a vacuum-driven process, or inert-driven process, we mean reactions which endin a state with vacancies only.

This implies from Eq. (2.12)w1,0(1, 0) = w0,1(0, 1) = w1,1(1, 1) = 0(4.2)This means that there is no production of pairs of A particles, in other words theprocess ∅+ ∅→A + A is forbidden.

Sincew1,0(α, β) = w0,1(α, β) = 0(4.3)because of the Z2 symmetry, we are left only with the processesa.) annihilation, with the rate w1,1(0, 0)A + A →∅+ ∅(4.4)b.) diffusion to the right, with the rate w1,1(0, 1)A + ∅→∅+ A(4.5)c.) diffusion to the left, with the rate w1,1(1, 0)∅+ A →A + ∅(4.6)We use now Eqs.

(3.4, 3.5, 3.6), take into account thatE00F = E01 , E11F = E10 , σz = E00 −E11,(4.7)11

and findTi=ϕ(i)(1, 1)ϕ(i)(0, 0)w1,1(0, 0)E01 ⊗E01+ϕ(i)(0, 1)ϕ(i)(1, 0)w1,1(1, 0)E10 ⊗E01 + ϕ(i)(1, 0)ϕ(i)(0, 1)w1,1(0, 1)E01 ⊗E10Ui=14 (w1,1(0, 1) + w1,1(1, 0) + w1,1(0, 0)) 1 ⊗1+14 (w1,1(0, 0) −w1,1(0, 1) −w1,1(1, 0)) σz ⊗σz−14w1,1(0, 0) (σz ⊗1 + 1 ⊗σz)+14 (w1,1(1, 0) −w1,1(0, 1)) (σz ⊗1 −1 ⊗σz)(4.8)Up to this point we still have freedom in the choice of the function ϕ(k)(βk, βk+1) inEq. (2.8).

We now takeϕ(k)(1, 0)ϕ(k)(0, 1) =vuutw1,1(1, 0)w1,1(0, 1) = q ; k = 1, 2, ..., L −1,(4.9)and it is clear that q is real since the rates are real. From Eqs.

(4.9) and (2.11) wegeth(k)(1)h(k)(0) = q1−kλ−1,(4.10)where λ is an arbitrary parameter. The diffusion constant D, expressed asD =qw1,1(0, 1)w1,1(1, 0),(4.11)fixes the time scale.

We then finally obtainH = D (H0 + H1)(4.12)whereH0=−12L−1Xi=1hσxi σxi+1 + σyi σyi+1 + ∆σzi σzi+1 + (1 −∆′)σzi + σzi+1−12q −q−1 σzi −σzi+1+ 2∆′ −∆−2(4.13)H1=−w1,1(0, 0)λ2DL−1Xi=1q1−2iE01i E01i+1(4.14)∆= q + q−12−w1,1(0, 0)2D, ∆′ = 1 −w1,1(0, 0)2D(4.15)12

Let us now discuss the structure of this Hamiltonian. H0 is hermitian, U(1) symmetricand its properties are going to be discussed shortly.

H1 is non-hermitian and has onlya Z2 symmetry, corresponding to the transformationsσxi →−σxi, σyi →−σyi , σzi →σzi(4.16)which makes H only Z2-symmetric (as expected) and non-hermitian.Let us discuss the case of pure diffusion, that is w1,1(0, 0) = 0. We have H1 = 0andH0 = −12L−1Xi=1"σxi σxi+1 + σyi σyi+1 + q + q−12σzi σzi+1 −q −q−12σzi −σzi+1−q + q−12#(4.17)This Hamiltonian is the UqSU(2) symmetric Hamiltonian of Pasquier and Saleur [26]where the deformation parameter q has a clear physical interpretation, see Eq.

(4.9).Moreover, as shown in Appendix C, H0 is the sum of generators of a given quotient ofa Hecke algebra (called a Temperley-Lieb algebra). As discussed in detail in AppendixC, for any number of species, asymmetric diffusion defines various quotients of Heckealgebras (we think this is an important observation).

As a corollary one can showthat for any number of species asymmetric diffusion is always massive.We now consider the case w1,1(0, 0) ̸= 0, that is the full nonhermitian HamiltonianEq. (4.13) which has a special property.

From the expression (4.15) of H1 we seethat one can take the factor λ2 arbitrary without changing the spectrum. This iscertainly so, since this factor only serves to parametrize a similarity transformation.On the other hand, one can also see by direct inspection of the matrix elements of Hthat the characteristic polynomial of H is not affected by the existence of H1.

Let usbriefly present the argument since we are going to use this repeatedly later on. If Cand D are square matrices and if X is a rectangular matrix, it is well known thatdet CX0D!= det C det D(4.18)Now, H0 has a U(1) symmetry and has thus a block-diagonal form.The corre-sponding quantum number labelling the blocks is the number of A particles present.Acting with H1 reduces this quantum number by two and this plays the role of X inEq.

(4.18). Thus the above determinant formula can be applied and the independenceof the spectrum of H from the operator H1 follows.Let us choose w1,1(0, 0) such that ∆= 0.

This corresponds accidentally to thechoice of all the previous numerical simulation studies of the one-species annihilationprocess where one took q = 1 and w1,1(0, 0) = 2D. In this case H can be studiedin terms of free fermions.

This will be done in Section 5 where some new resultsare presented. In particular we find that for q = 1, H0 can be written in terms offermionic number operators as followsH0 ∼Xk kL!2a+k ak(4.19)13

which describes a Pokrovsky-Talapov phase transition [27].Notice the quadraticdispersion relation which is expected in any diffusion problem.In Appendix B we consider a complementary point of view which is related to theproblem of integrability of quantum chains with the structure Eq. (4.12) in which H0is integrable (in the present case it corresponds to a six-vertex model) and H1 doesnot affect the spectrum of H. We show that H is related to a seven-vertex model forwhich the Yang-Baxter equations are valid and hence H is integrable.We now consider the case ∆̸= 0.This brings us, see Eq.

(4.13), to discussproperties of the HamiltonianH′ = −12L−1Xi=1hσxi σxi+1 + σyi σyi+1 + ∆σzi σzi+1 + hσzi + σzi+1i(4.20)This Hamiltonian is integrable [28] and its phase diagram is shown in Fig. 1.

Forh = 0, the system is massive with a ferromagnetic ground state if ∆> 1, massless andconformally invariant if −1 ≤∆≤1 and again massive with an antiferromagneticground state if ∆< −1.For a given h, the system is massive ferromagnetic if∆> 1 −h, then undergoes a Pokrovsky-Talapov transition at ∆P T = 1 −h, is in amassless incommensurate phase for ∆c < ∆< 1 −h and reaches again the massiveantiferromagnetic phase if ∆< ∆c.It can be shown [29] that in the continuum, along the Pokrovsky-Talapov line,the spectrum of H′ is, up to normalisation, given by Eq. (4.19) for any ∆.

In otherwords, the system is massless with a quadratic dispersion in the momentum k.With this knowledge in hand, let us discuss some properties of H0. Since ∆′ < ∆,we have h > 1 −∆and the system is massive.

If however, q = 1, we have ∆′ = ∆and we are on the Pokrovsky-Talapov line where the system is massless. There aresome immediate questions to ask about this system.

If we defineǫ = q + q−12−1(4.21)and if τ denotes the relaxation time, we are interested in the exponent ητ ∼ǫ−η(4.22)It can be shown that η = 1 is independent of ∆using the standard lore of Heckealgebras, see Appendix C. This result is to be expected since we can change the valueof ∆by changing D, see Eq. (4.15).

However, changing D merely changes the timescale. More generally, the full finite-size scaling form should readτ = Lη1F(x) , x = ǫLη2(4.23)in the simultaneous limit ǫ →0, L →∞with x kept fixed.

Concerning the concen-tration of A particlescA = 1LLXi=1E11i(4.24)14

in the critical regime (q = 1), we are interested in< cA > (t) = Lxφ(z) , z = tL−2(4.25)for both t and L being large with z fixed. The exponent 2 in the definition of zwas taken because of the quadratic dispersion relation Eq.

(4.19). In Section 5 weshall find x = −1.

Similarly, correlation functions can be introduced and calculated.We shall present the analysis in Section 5 for the case ∆= 0. For the case ∆̸= 0,although we expect on physical grounds the same results, the explicit proof is stillmissing.

The Bethe ansatz equations for the wave equations are known only for H0but not for H. In Appendix A it is shown how in principle the knowledge of theeigenfunctions of H0 can help to find the eigenfunctions of H. In a different approachone could start with the seven-vertex model (which is not limited to the case ∆= 0),and perform the Bethe ansatz there. The wave functions thus found could be usedfor H. Whether this whole program is manageable remains to be seen.4.2Z2 non-invariant, parity invariant vacuum-driven pro-cessesFrom now on, we take always the left-right symmetric case, see Eq.

(2.6), in allreaction rates, in particular w1,1(0, 1) = w1,1(1, 0) = D. We choose units of time suchthat D = 1. To the processes studied before we now add the followingd.) coagulation, with rate w1,0(0, 1)A + A →∅+ A(4.26)e.) decoagulation, with rate w1,0(1, 1)∅+ A →A + A(4.27)f.) death, with rate w1,0(0, 0)A + ∅→∅+ ∅(4.28)We take ϕ(i)(0, 1) = ϕ(i)(1, 0)(i = 1, 2, ..., L −1) and getTi=E01 ⊗E10 + E10 ⊗E01+ ϕ(i)(1, 1)ϕ(i)(0, 0)w1,1(0, 0)E01 ⊗E01+ϕ(i)(1, 0)ϕ(i)(0, 0)w1,0(0, 0)E01 ⊗E00 + E00 ⊗E01+ϕ(i)(1, 1)ϕ(i)(1, 0)w1,0(0, 1)E01 ⊗E11 + E11 ⊗E01+ϕ(i)(1, 0)ϕ(i)(1, 1)w1,0(1, 1)E10 ⊗E11 + E11 ⊗E10(4.29)15

Ui=12w0,0(1, 0) + 14w0,0(1, 1) + w0,0(1, 1)4−w0,0(1, 0)2!σz ⊗σz−w0,0(1, 1)4(1 ⊗σz + σz ⊗1)(4.30)wherew0,0(1, 1)=2w1,0(0, 1) + w1,1(0, 0)w0,0(1, 0)=w1,0(0, 0) + w1,0(1, 1) + 1(4.31)Thus the Hamiltonian depends on four parameters. Two cases have to be consideredseparately.

In the first case we have only coagulation, in the second case we havecoagulation and decoagulation. If we have no decoagulation, w1,0(1, 1) = 0.

ThenH = H0 + H1(4.32)whereH0=−12L−1Xi=1hσxi σxi+1 + σyi σyi+1 + ∆σzi σzi+1 + (1 −∆′)σzi + σzi+1+ 2∆′ −∆−2i(4.33)H1=−L−1Xi=1"ϕ(i)(1, 1)ϕ(i)(0, 0)w1,1(0, 0)E01i E01i+1 + ϕ(i)(1, 0)ϕ(i)(0, 0)w1,0(0, 0)E01i E00i+1 + E00i E01i+1+ ϕ(i)(1, 1)ϕ(i)(1, 0)w1,0(0, 1)E01i E11i+1 + E11i E01i+1#(4.34)∆′=1 −w1,0(0, 1) −12w1,1(0, 0)(4.35)∆=∆′ + w1,0(0, 0)(4.36)We have checked using the same arguments as above that the spectrum of H isindependent of the presence of H1. Since ∆′ < ∆, we are in a massive phase.

Thesystem is massless, i.e. ∆= ∆′, if w1,0(0, 0) = 0, corresponding to the absence ofdeath processes.

A straightforward calculation shows that the relaxation time τ scaleslikeτ ∼ w1,0(0, 1)w1,1(1, 0)!−1(4.37)This can be seen by repeating the same arguments as for Eq. (4.22).

The resultEq. (4.37) can be understood easily if one keeps in mind the time evolution of thesystem.

If one considers the later stages of the process when few “molecules” are left,few of them will meet and annihilate. On the other hand, any one of them can “die”since this is an individual process.16

In the absence of the death process some results are already known for the pureannihilation process [12] or the coagulation process [11] and it remains to be seenhow much more can be learnt using the quantum chain formulation.We now consider the case when the coagulation and the decoagulation processescoexist and we have all terms in Eq. (4.29).

Then little can be said. One interestingcase [30] is when apart from just diffusion, only coagulation and decoagulation arepresent, that is w1,1(0, 0) = w1,0(0, 0) = 0.

We chooseϕ(i)(1, 1)ϕ(i)(1, 0) =vuutw1,0(1, 1)w1,0(0, 1) , (i = 1, 2, ..., L −1). (4.38)We stress that this transformation is singular when the decoagulation rate w1,0(1, 1)goes to zero.

We get a hermitian Hamiltonian H = H0 + H1 with H0 being given byEq. (4.33) with∆′=1 −w1,0(0, 1)(4.39)∆=∆′ + w1,0(1, 1)(4.40)andH1 = −L−1Xi=1q(∆−∆′)(1 −∆′)σxi E11i+1 + E11i σxi+1(4.41)where ∆′ < ∆and again we suspect a massive phase, at least when ∆′ is close to ∆.Note that now H1 is hermitian and its presence does change the spectrum of H. Thischange is in fact quite important.

If we look directly at the master equation, besidesthe trivial stationary probability distribution Eq. (2.13) there is a second one whichsatisfies ∂tP = 0 and is given byP({β}; t) =LYk=1p(βk)(4.42)wherep(β)=expµ(−1)β + νµ=lnvuutw1,0(0, 1)w1,0(1, 1) , ν = −µ −ln 1 + w1,0(1, 1)w1,0(0, 1)!

(4.43)The probability distribution Eq. (4.42) corresponds to a one-dimensional Ising modeldefined on the dual lattice and the site variable βk corresponds to a link variable ofthe dual lattice, see also Section 8.The existence of two stationary probability distributions satisfying ∂tP = 0 makesthe Hamiltonian H have a degenerate vacuum.The existence of a decoagulationprocess, no matter how small it is, implies, if the annihilation process is absent,that the stationary configuration should correspond to Eq.

(4.42), unless the initial17

state is the empty lattice. If we now turn on the annihilation mechanism, the finalconfiguration will be the empty lattice.

Since we always have massive behaviour thistransition between stationary states will correspond to a first-order nonequilibriumphase transition which may be accounted by the fact that no symmetry of H isbroken. This can now be studied by performing perturbative calculations in the limitof a small decoagulation rate.We are aware of the following fascinating puzzle.Although the Hamiltoniandescribing the coagulation and decoagulation processes is not known to be integrableit was shown in [30] that the gap-probability function for this model can be computedexactly and this is an indication that the model is integrable.

This is at least the case ifone considers the creation-annihilation model with diffusion which will be discussednext (recall that Glauber’s solution [19] was discovered without using the Jordan-Wigner transformation). We will return to this problem in a future publication (seealso Section 6).4.3Z2 and parity invariant, not vacuum-driven processesThe full master equation without any restrictions includes besides the processes con-sidered so far the following two, see Eq.

(4.2)g.) creation, with rate w1,1(1, 1)∅+ ∅→A + A(4.44)h.) birth, with rate w1,0(1, 0)∅+ ∅→A + ∅(4.45)From probability conservation, we havew0,0(0, 0)=w1,1(1, 1) + 2w1,0(1, 0)w0,0(1, 0)=w1,0(0, 0) + w1,0(1, 1) + w1,1(1, 0)(4.46)w0,0(1, 1)=2w1,0(0, 1) + w1,1(0, 0)The Hamitonian isH=L−1Xi=114 (w0,0(0, 0) + 2w0,0(1, 0) + w0,0(1, 1))+14 (w0,0(0, 0) + w0,0(1, 1) −2w0,0(1, 0)) σzi σzi+1+14 (w0,0(0, 0) −w0,0(1, 1))σzi + σzi+1−w1,1(1, 0)E01i E10i+1 + E10i E01i+1−λ2w1,1(0, 0)E01i E01i+1 −λ−2w1,1(1, 1)E10i E10i+118

−λw1,0(0, 0)E01i E00i+1 + E00i E01i+1−λ−1w1,0(1, 0)E10i E00i+1 + E00i E10i+1−λw1,0(0, 1)E01i E11i+1 + E11i E01i+1−λ−1w1,0(1, 1)E10i E11i+1 + E11i E10i+1i,(4.47)where we have choosen h(k)(1)h(k)(0) = λ, (k = 1, 2, ..., L) in Eq. (2.11).

This Hamiltonianhas an interesting property. It can always be brought to a hermitian form througha similarity transformation if one out of the three possible forward-backward reac-tions (annihilation-creation, death-birth, coagulation-decoagulation) take place.

If,however, the following relation between the rates is satisfiedw1,1(1, 1)w1,1(0, 0) = w1,0(1, 1)w1,0(0, 1)!2= w1,0(1, 0)w1,0(0, 0)!2(4.48)H can be made hermitian even when all three pairs of reactions occur. The physicalsignificance of Eq.

(4.48) has still to be explored.We shall confine ourselves to the Z2 preserving processes which include besidesdiffusion only annihilation and creation processes. We make the transformationϕ(k)(1, 1)ϕ(k)(0, 0) =vuutw1,1(1, 1)w1,1(0, 0) , (k = 1, 2, ..., L −1),(4.49)and find H/w1,1(1, 0) = H0 + H1 whereH0=−12L−1Xi=1hσxi σxi+1 + σyi σyi+1 + ∆σzi σzi+1 + (1 −∆′)σzi + σzi+1+ ∆−2iH1=−12L−1Xi=1q(∆′ −∆)(2 −∆−∆′)σxi σxi+1 −σyi σyi+1(4.50)∆′=1 −w1,1(0, 0) −w1,1(1, 1)2w1,1(1, 0)(4.51)∆=∆′ −w1,1(1, 1)w1,1(1, 0)(4.52)This is just the Hamiltonian corresponding to the kinetic Ising model with purely re-laxational dynamics [31] and H is hermitian.

The limit when the elements of H1 aresmall does need some care. If the annihilation and creation rates are equal, ∆′ = 1which allows us to do perturbation theory in the coupling 1 −∆of Eq.

(4.50). How-ever, if the rates are not equal and one just takes w1,1(1, 1) to zero, the limit for theeigenvectors is singular as is the transformation Eq.

(4.49), although the eigenvalue19

spectrum is not affected. Therefore, the heuristically appealing relationship [32] be-tween the kinetic Ising model with Glauber dynamics and the annihilation processshould be considered with care if one works (like in Ref.

[31]) with the hermitianformulation given by Eq. (4.50).5Exact solution of a two-state systemIn this section, we shall discuss the dynamics of the two-states system describedby the Hamiltonian Eq.

(4.12) in the free fermion case ∆= 0. (For a connexionbetween this Hamiltonian and a seven-vertex model, see Appendix B.) This modeldescribes A particles diffusing and annihilating in pairs.

The motivation for treatingthis particular case in great detail is twofold.Firstly, we would like to show that a finite-size scaling theory can be formulated fornonequilibrium systems. This point is particularly important if (in contradistinctionto this example) the problem is not exactly solvable.

In that case, one will have tosolve the problem numerically for finite chains and then extrapolate the results toinfinite systems, as one does for equilibrium problems [33] (see also Appendix A).One expects that the finite-size scaling proven for this model is of general validity.For instance, the concentration of particles per site < n > /L obeys the relationc = Lxφ(z)(5.1)where z = 4Dt/L2 and φ(z) ∼z−α as z →0. Moreover, x is related to the large timebehaviour of c ∼t−α, namely x = −2α.Secondly, we want to stress the interest of the (connected) two-point functionG(r, t) =< nmnm+r > −< nm >< nm+r >(5.2)where m indicates the position of a lattice site.

Here, we shall pose two differentquestions about G(r, t). First, we ask for the large time behaviour (t →∞) if r iskept fixed and look for the exponent vrG(r, t) ∼tvr(5.3)Secondly, we consider the scaling limit where simultaneously r, t →∞such thatu = r2/t is fixed and define a critical exponent yG(r, t) ∼r−yg(u)(5.4)where g is a scaling function.We believe that for nonequilibrium problems theexponents y’s might turn out to be as fundamental as the critical exponents of two-dimensional equilibrium statistical mechanics, since the information given by theinitial probability function should be hidden in the scaling function g(u) and conse-quently the y′s should be universal (see also Ref.

[32]).20

Finally, we would like to present powerful techniques which allow us to reducethe calculation of nonequilibrium averages, see eq. (3.18), to vacuum expectationvalues, thereby extending the approach initiated by Lushnikov [34] to the calculationof correlation functions.The quantum Hamiltonian H (4.12) contains the parameterǫ = 12q + q−1−1 ≥0(5.5)where ǫ measures the left-right asymmetry of the diffusion process.

In the remainderof this section we take ǫ = 0. We do not make use of the change of variables givenby the function ϕ(i)(α, β) described in the last section.

In terms of Pauli matricesσ± = (σx ± iσy) /2, the Hamiltonian is equivalent toH = −DLXn=1hσ+n σ−n+1 + σ−n σ+n+1 −2σ+n σ−n + 2σ−n σ−n+1i(5.6)where D is the diffusion constant. In this section for simplicity we only use periodicboundary conditions.5.1Diagonalization and the generating functionThe diagonalization H is fairly standard.

By restricting ourselves to the case L = 2Meven we define fermionic variables via the Jordan-Wigner transformationσ−n=(−1)Pm

. ., Mqodd=2k2M π , k = 0, 1, .

. .

, M(5.10)21

if the total number of fermions N = PLm=1 a+mam is an even or odd number, respec-tively. The Hilbert space associated to Eq.

(5.9) can therefore be splitted into twodisjoints sectors depending on the parity of the fermion number. The sector with Nodd (N even) will be related to the case where the ground state has a single particle(no particle).Looking again at Eq.

(5.9), we note that H = P0≤q≤π Hq has block-diagonalstructure. The time-evolution equation for wave functions ∂tΦ = −HΦ can be brokenup into separate equations for each blockΦ(t) =Y0≤q≤πΦq(t) , ∂tΦq = −HqΦq; (0 ≤q ≤π).

(5.11)Each block is generated by acting with ˜a+q and ˜a+−q on the state |vac⟩without fermions,i.e. ˜aq |vac⟩= 0.

If q ̸= 0, π, Hq can be written as a 4 × 4 matrix. In the basis|0⟩= |vac⟩, |1⟩= ˜a+q |vac⟩, |2⟩= ˜a+−q |vac⟩and |3⟩= ˜a+q ˜a+−q |vac⟩it takes the formHq = −2D0−2 sin qcos q −1cos q −12 cos q −2(5.12)while for q = 0, π, the blocks H0 and Hπ are already diagonal.

The general solutionof Eq. (5.11) corresponds toΦq(t) =αq(t)˜a+q ˜a+−q + γq(t)˜a+q + δq(t)˜a+−q + βq(t)|vac⟩(5.13)where the functions α, β, γ, δ satisfy the differential equations˙αq(t)=4D (cos q −1) αq(t)˙βq(t)=−4D sin q αq(t)˙γq(t)=2D (cos q −1) γq(t)˙δq(t)=2D (cos q −1) δq(t)(5.14)The solutions of these equations are promptly derivedαq(t)=αq(0) exp [−4Dt (1 −cos q)]βq(t)=βq(0) + cot q2 {αq(t) −αq(0)}γq(t)=γq(0) exp [−2Dt (1 −cos q)]δq(t)=δq(0) exp [−2Dt (1 −cos q)](5.15)For an arbitrary initial condition the general solution will be given by linear combina-tions of the wave functions Eq.

(5.11). A general study considering arbitrary generalinitial conditions is not straightforward and is not in our intentions.

Since our maininterest here is the finite-size scaling behaviour of the system, we will hereafter choose22

the simple initial condition where we have no vacancies present. This condition wasused by Lushnikov [34] and corresponds to the situations where we have a single wavefunction (5.11), i.e.

ψ(t) = Φ(t) withαq(0) = −1 , βq(0) = γq(0) = δq(0) = 0,(5.16)where q takes the values qeven in the set (5.10), since in this case the fermion numberN is even.In order to calculate the concentration of particles and correlations, following thework of Lushnikov [34] it is convenient to introduce generating functions. Since (seeSection 3) the wave function takes the role of the probability distribution introducedearlier, a generating function convenient for our purpose is defined byF({z}, t) = F(z1, .

. .

, zL, t) =P{C}QLi=1 znii ψ({C}; t)P{C} ψ({C}; t)(5.17)where ni = 0, 1 and zi denote the occupation number and fugacity at the site i, Cis a configuration {n1, . .

. , nL} of occupied or empty sites.

In Eq. (5.17) we takeinto account that ψ does not, in general, satisfy the normalization condition of aprobability distribution.

Using the results of Section 3, in particular (3.14), we canbring the calculation of F({z}, t) to an equilibrium problem, i.e.F({z}, t)=F0(t)X{C}LYi=1znii ψ({C}; t)(5.18)=F0(t) ⟨vac| exp LXℓ=1zℓσ−(ℓ)! Yq>0αq(t)˜a+q ˜a+−q + βq(t)|vac⟩(5.19)where we have already used the initial condition Eq.

(5.16) and F0(t) is determinedfrom the normalization condition F(1, . .

., 1, t) = 1 for all t. In order to understandEq. (5.19), it is useful to consider the identityexp LXm=1zmσ−(m)!=LXk=0Xm1>m2>...>mkzm1σ−(m1) · · ·zmkσ−(mk)(5.20)which we can derive by expanding the exponentials and using the relations (σ−(m))2 =0 and [σ−(m), σ−(m′)] = 0.

It is now clear that ⟨vac| expPLℓ=1 zℓσ−(ℓ)correspondsto the state ⟨0| of Section 3.In order to calculate F({z}, t) we observe that due to the ordering m1 > m2 >. .

. > mk we get⟨vac| σ−(m1)σ−(m2) .

. .σ−(mk) = ⟨vac| am1am2 .

. .

amk(5.21)On the other hand, we also have from Eq. (5.8)˜a+q ˜a+−q = 2LXmXn>msin(q(n −m)) a+n a+m(5.22)23

Because of the product structure Eqs. (5.11, 5.13) of φ, the generating function canalso be written as F =Qq>0 Fq.

Each factor contains only two fermionic creationoperators at most. So we only need to calculateXm′Xm>m′⟨vac| σ−(m)σ−(m′)˜a+q ˜a+−q |vac⟩=2LXn′,m′Xm>m′Xn>n′hsin(q(n −n′)) ⟨vac| amam′a+n a+n′ |vac⟩i=2LXn′,m′Xm>m′Xn>n′[sin(q(n −n′)) (δn,m′δn′,m −δn,mδn′,m′)]=−2LXn′Xn>n′sin(q(n −n′))(5.23)In the sequel, we shall need two identities, which are obtained using Eq.

(5.10), withm fixedXn>msin(q(n −m))=12"cot q2 + cos(q(m −1/2))sin(q/2)#Xn0 αq(t) · −2LXmXn>mznzm sin(q(n −m))!+ βq(t)!

(5.25)We fix F0(t) from the condition F(1, . .

., 1, t) = 1 for all t. The final result for thegenerating function isF({z}, t) =Yq>0αq(t) ·−2LPmPn>m znzm sin(q(n −m))+ βq(t)−αq(t) cot q2 + βq(t)(5.26)5.2Calculation of the mean concentrationThe mean number of particles at an arbitrary site m for a chain of length L is nowobtained from< nm >= zm∂∂zmF({z}, t)z1=...=zL=1= L−1 Xq>0 −2αq(t) cot q2−αq(t) cot q2 + βq(t)! (5.27)Due to the translational invariance of the particular initial state considered here(Eq.

(5.16)) the concentration of particles per site is simplyc =< nm >(5.28)Substituting Eqs. (5.13, 5.16) into (5.27) we obtainc(t) = 2L−1 Xq>0exp (−4Dt (1 −cos q))(5.29)24

5.3Large-time behaviour and finite-size scalingWe now discuss the late stages dependence of the concentration c(t). In the limitL →∞for t fixedc(t) = 1πZ π0 e−4Dt(1−cos q)dq = e−4DtI0(4Dt)(5.30)where I0(x) is a modified Bessel function.

For Dt ≫1 one getsc(t) ≃(8πDt)−1/2(5.31)Let us now analyse the regime of finite-size scaling where L →∞, t →∞buttL−2 fixed. In this case we obtainc=2L∞Xk=1exp −2π2zk −122!

(5.32)=1LΘ2(0, 2πiz)(5.33)=1Ls12πz 1 + 2∞Xℓ=1(−1)ℓexp−ℓ2/(2z)! (5.34)where Θ2(z, τ) is a Jacobi theta function andz = 4DtL−2(5.35)is the finite-size scaling variable.

Eq. (5.34) is obtained from (5.32) by using thePoisson resummation formula.

From Eqs. (5.32-5.34) and (5.1) we get the criticalexponent x = −1.5.4Fluctuations around the mean valuesNext, we briefly discuss the fluctuations around < nm >.

For simplicity, we takez1 = . .

. = zL = z in Eq.

(5.17). Then the generating function F = F(z, t) and wehave< n2m >=1L2 z ∂∂z!2F(z, t)z=1=< nm >2 +2 < nm > /L −1L2Xq>0 −2αq(t) cot q2−αq(t) cot q2 + βq(t)!2(5.36)We get< n2m > −< nm >2=2/Le−4DtI0(4Dt) −e−8DtI0(8Dt)≃1L1√2πDt 1 −1√2!

(5.37)25

as t →∞. This means that for fixed, but large t< δn2m > / < nm >2∼t1/2L−1(5.38)which accounts for the fact that as the time grows the annihilation of couples ofparticles induces larger fluctuations in the particle concentration.5.5Correlation functionsWe now turn towards the calculation of correlation functions.

As an example whichillustrates the general technique, we calculate the connected correlation function de-fined in Eq. (5.2).

From the definition of the generating function (5.17) we can writeG(r, t) =< nmnm+r > −< nm >2=∂2∂zm∂zm+rln F({z}, t)z1=...=zL=1(5.39)A straightforward calculation gives usG(r, t) = 2LXq>0 −sin(qr) αq(t)−αq(t) cot q2 + βq(t)!−4L2Xq>0 −αq(t) cot q2−αq(t) cot q2 + βq(t)!2(5.40)which does not depend on m due to the fact that the Hamiltonian Eq. (5.6) as wellas the initial probability distribution are translational invariant.Analysing the relative importance of the two terms contributing to G(r, t), we seethat in the limit L →∞, the first term is of order unity, while the second one is oforder O(L−1).This correlation can be evaluated in the same way as we did for the concentration.Let us consider initially the case of the correlation of next-neighbour particles.

WefindG(1, t)=2LXq>0(1 −cos q) exp [−4Dt (1 −cos q)]=e−4Dt [I0(4Dt) −I1(4Dt)]≃π(8πDt)−3/2(5.41)as t →∞. The same result can be obtained using Glauber’s dynamics for the Isingmodel at T = 0 (see also Section 8) and considering domain walls two-point functions[35].

The initial condition we are considering in the diffusion-annihilation problem(no vacancies) corresponds to a fully ordered antiferromagnetic state in the kineticIsing model. On the other hand, for arbitrary but finite r, we obtain analogouslyG(r, t) ≃rG(1, t)(5.42)This result reflects the fact that at late times it is more difficult to find particles closeto each other than far apart.

From Eq. (5.3) this gives us the exponents vr = −3/2.26

Finally, in connection with Eq. (5.4), let us consider the limit where r →∞,t →∞but u = r2/t stays finite.

This leads toG ≃π(8πD)−3/2r−2u3/2e−u/(2D)(5.43)and we read offthe exponent y = −2 and recognize the scaling function g(u) =π(8πD)−3/2u3/2e−u/(2D), see Eq. (5.4).

We remark that if we consider the domainwalls two-point function in the zero-temperature kinetic Ising model with Glauberdynamics [35] for an initial probability distribution with zero magnetization (whichis a different initial condition than that we are considering here), we obtain the sameresult. This is an indication in favour of our conjecture that the y exponents areuniversal.6Three-state models with Z3 symmetry and va-cuum-driven processesAs noticed in Section 4, behind many processes in the two-states models there is theXXZ model in a magnetic field.

We shall now show that there are several integrablequantum chains which play a similar role for three-states models with Z3 symmetry.We shall restrict ourselves to the Z3-symmetric case for simplicity.This alreadycontains the two-species annihilation process but not the Z2 symmetric trappingreaction to which we hope to return in a future publication.Let 0 stand for avacancy, 1 for the particle A and 2 for the particle B.We use the general results of Section 3 and the conditions Eqs. (2.5,2.12) andderive the HamiltonianH = U −T(6.1)T=L−1Xi=1hw2,1(0, 0)E02i E01i+1 + w1,2(0, 0)E01i E02i+1+w2,1(1, 0)E10i E01i+1 + w1,2(0, 1)E01i E10i+1+w2,1(2, 1)E21i E12i+1 + w1,2(1, 2)E12i E21i+1+w2,1(0, 2)E02i E20i+1 + w1,2(2, 0)E20i E02i+1+w2,1(1, 1)E10i E12i+1 + w1,2(0, 2)E01i E21i+1+w2,1(2, 0)E21i E01i+1 + w1,2(1, 1)E12i E10i+1+w2,1(0, 1)E02i E12i+1 + w1,2(2, 2)E20i E21i+1+w2,1(2, 2)E21i E20i+1 + w1,2(1, 0)E12i E02i+1+ w2,1(1, 2)E10i E20i+1 + w1,2(2, 1)E20i E10i+1i(6.2)U=L−1Xi=1h(w2,1(2, 0) + w1,2(0, 2)) E11i E11i+1 + (w2,1(0, 1) + w1,2(1, 0)) E22i E22i+127

+ (w2,1(1, 0) + w1,2(2, 2)) E00i E11i+1 + (w2,1(2, 2) + w1,2(0, 1)) E11i E00i+1+ (w2,1(1, 1) + w1,2(2, 0)) E00i E22i+1 + (w2,1(0, 2) + w1,2(1, 1)) E22i E00i+1+ (w2,1(2, 1) + w1,2(0, 0)) E11i E22i+1 + (w2,1(0, 0) + w1,2(1, 2)) E22i E11i+1+ (w2,1(1, 2) + w1,2(2, 1)) E00i E00i+1i. (6.3)Notice that for the time being we do not use any equivalence transformation (2.8) aswe did in Sec.

(4).We first consider the case of asymmetric diffusion∅+ A →A + ∅with rate w2,1(1, 0)A + B →B + Awith rate w2,1(2, 1)B + ∅→∅+ Bwith rate w2,1(0, 2)A + ∅→∅+ Awith rate w1,2(0, 1)B + A →A + Bwith rate w1,2(1, 2)∅+ B →B + ∅with rate w1,2(2, 0)(6.4)and let us assume that we havew2,1(1, 0)=w1,2(2, 0) = w2,1(2, 1) = ΓLw1,2(0, 1)=w2,1(0, 2) = w1,2(1, 2) = ΓR. (6.5)By definingq =sΓLΓR(6.6)we now obtainH=DL−1Xi=1q + q−12−qXb>aEabi Ebai+1 + q−1 Xb

Through a similarity transformation S,we can bring this Hamiltonian to the formH′ = SHS−1=DL−1Xi=1q + q−12−Xa̸=bEabi Ebai+1 + q + q−122Xa=0Eaai Eaai+1+q −q−12Xa̸=bsign(a −b)Eaai Ebbi+1(6.8)This similarity transformation does not have the simple local form of Eq. (2.8).

Theargument goes as follows. We first notice that the Hamiltonian H′ given by Eq.

(6.8) isthe exactly integrable UqSU(3) chain [36] and corresponds to the anisotropic version28

of the spin 1 model introduced by Sutherland [37], (see also Appendix C, where theconnexion with the Hecke algebra is shown). On the other hand H′ can be broughtby an equivalent transformation to the four-parameter deformation of the SU(3)symmetric chain [38]H′′=˜SH′ ˜S−1 = D(L−1Xi=1"q + q−12−f10E10i E01i+1 + f −110 E01i E10i+1+f20E20i E02i+1 + f −120 E02i E20i+1 + f21E21i E12i+1 + f −121 E12i E21i+1+q + q−122Xa=0Eaai Eaai+1 + q −q−12Xa̸=bsign(a −b)Eaai Ebbi+1(6.9)We observe that if in Eq.

(6.9) we take f21 = f31 = f32 = q we recover Eq. (6.8).The equivalence between the chain given by Eq.

( 6.7) and the UqSU(3) chain (6.8)is similar to the two-state model in the case of an asymmetric definition, where wegot UqSU(2). The chain Eq.

(6.8) is massive unless q = 1, which corresponds to theleft-right symmetric case and the physical interpretation for these results is similarto the two species case of Section 4. The case where the process A + B →B + Ais suppressed is, as shown in Appendix C, again related to a quotient of a Heckealgebra and the eigenvalues of the Hamiltonian are the same (the degeneracies maybe different) as for the UqSU(2) symmetric one given by Eq.

(4.17).We shall take from now on q = 1 and parity conservation. We assume that thediffusion rates arew2,1(1, 0)=w1,2(0, 1) = w2,1(0, 2) = w1,2(2, 0) = Dw2,1(2, 1)=w1,2(1, 2) = λD(6.10)We consider now the annihilation processesA + B →∅+ ∅with rate w1,2(0, 0) = w2,1(0, 0) = αABDA + A →∅+ Bwith rate w1,2(0, 2) = w2,1(2, 0) = αAADB + B →∅+ Awith rate w1,2(1, 0) = w2,1(0, 1) = αBBD(6.11)In this case, the Hamiltonian has a special formH = D(H0 + H1)(6.12)whereH0=L−1Xi=1h2αAAE11i E11i+1 + 2αBBE22i E22i+1+(λ + αAB)E11i E22i+1 + E22i E11i+1+E00iE11i+1 + E22i+1+E11i + E22iE00i+1−E10i E01i+1 −E01i E10i+1 −E20i E02i+1 −E02i E20i+129

−λE21i E12i+1 + E12i E21i+1i(6.13)H1 = −L−1Xi=1hαABE01i E02i+1 + E02i E01i+1+αAAE21i E01i+1 + E01i E21i+1+ αBBE02i E12i+1 + E12i E02i+1i(6.14)Using the arguments from Section 4, one can check that the spectrum of the hermitianoperator H0 coincides with the one of H/D. This Hamiltonian describes the purereaction A + B →∅, if we takeλ = αAA = αBB = 0(6.15)Let us find cases where the Hamiltonian is integrable.First we look at theUqSU(3) chain Eq.

(6.8) with q = −1. If we add to the Hamitonian fields alongthe generators of the algebra, the system stays integrable since the supplementaryterms commute with the integrable UqSU(3) Hamiltonian.

We found only one casewhere this is possible. Takingλ = 1 , αAA = αAB = αBB = 2(6.16)we findH0 =L−1Xi=1−1 +Xa̸=bEabi Ebai+1 −2Xa=0Eaai Eaai+1 + 2σ0i + σ0i+1(6.17)The first three terms in Eq.

(6.17) give a SU(3) symmetric Hamiltonian, correspond-ing to q = −1 in Eq. (6.7).

The last term withσ0 =000010001(6.18)is an external field. This observation should allow us to study the critical propertiesof this chemical process.Another possibility occurs when λ = 0.

We can rewrite H0 in the following form,assuming αAA = αBBH0=L−1Xi=12αAA + αAB2−2σ0i σ0i+1 +2αAA −αAB2σzi σzi+1+ σ0i + σ0i+1 −τ +i τ −i+1 −τ −i τ +i+1 −ρ+i ρ−i+1 −ρ−i ρ+i+1i(6.19)where we have used the notationsτ + = E01 , τ −= E10 , ρ+ = E02 , ρ−= E20 , σz = E11 −E22(6.20)This Hamiltonian commutes with Σz and Σ0 whereΣz =LXi=1σzi , Σ0 =LXi=1σ0i(6.21)30

For the special choiceαAB = 2αAA(6.22)the Hamiltonian has the larger symmetry SU(2)⊗U(1) since H0 commutes also withΣ± which are defined byΣ+ =LXi=1σ+i, Σ−=LXi=1σ−i(6.23)using σ+ = E12 and σ−= E21.If we finally take αAA = 1, we haveH0 =L−1Xi=1hσ0i + σ0i+1−τ +i τ −i+1 −τ −i τ +i+1 −ρ+i ρ−i+1 −ρ−i ρ+i+1i(6.24)The chain Eq. (6.24) is integrable and massless as shown in Appendix C. It can alsobe related to t −J model [51] with J = 0.

Moreover the spectrum of H0 (not thedegeneracies) is that of a free fermion system. In Appendix C it is also shown thatnot only H0 but also H can be expressed in terms of a Hecke algebra which impliesthat the system is integrable.

This remains valid even if we don’t assume left-rightsymmetry. The quantum chain given by Eq.

(6.24) was discovered independently inRefs. [53] and [54].

In the first reference this chain was found as a special case in asearch of chains related to Hecke algebras. In the second reference it corresponds to aspecial case of integrable chains defined on nilpotent representations of the UqSU(2)quantum group with q = exp 2πi3 .

In the last reference the SU(2) ⊗U(1) symmetrywas observed. This triggered a further investigation by two of the authors to findthe whole symmetry of the chain and obtain its eigenspectrum [55].

We would liketo mention that for this system an exact result concerning the time-dependence ofthe total concentration < cA + cB > is already known [39]. Moreover, if we keepthe relation (6.22) but let the parameter αAA free, one can show that the spectrum(not the degeneracies) is the same as that of the XXZ model in a field, see Eq.

(4.20)with ∆= 1 −αAA and h = 1 −∆, which corresponds to the Prokrovski-Talapov lineof Fig. 1.

Up to now we have found several choices for the constants αAB and αAAfor which H0 is integrable. We failed to find others, although two other classes ofintegrable models, namely the chiral Potts [40] and the one with UqSU(2) , q3 = 1symmetry where periodic or semiperiodic representations [41] are taken, have therequired Z3 symmetry.Much work is still to be done on the phase structure of the three-state models.One of the questions to be asked is the connection between the case Eq.

(6.15), whereαAA = 0, which is known to be massless, to the case αAA = 1, which is massless aswell.Finally, we mention that in Appendix C (see Eq. (C.16)) we give an example of aZ2 symmetric three-state chain (only A + A →∅reaction) which is also related to aHecke algebra.31

7Conditions on the reaction rates for the exis-tence of steady statesSteady states are time-independent solutions of the master equation. In Section 2 wehave seen that if the reaction rates satisfy the condition Eq.

(2.12), then the emptylattice Eq. (2.13) is a steady state.

We now ask for the conditions on the rates suchthat the probability distributionP({β}) =LYk=1f(βk)(7.1)is a steady state, where f(β) is a given arbitrary function. Our motivation is notonly to understand better the structure of the ground states of the master equationand of the corresponding quantum Hamiltonians, but also to be able to addressthe question of the nonequilibrium behaviour of classical one-dimensional systems.This amounts to introduce a detailed balance condition for a prescribed equilibriumcondition P, while in the previous sections we have always taken the empty latticeas the equilibrium configuration.

Let us explain this point.Suppose that at temperature T we have a one-dimensional classical system definedby the probability functionP({β}) =LYk=1g(αk −αk+1)(7.2)If the β’s are site variables, the α’s are link variables.In our study we are going to consider the two-state model (Ising model) whereg2(βk) = g2(αk −αk+1) = exp(−1)βk/T(7.3)and the three-state model (chiral Potts model [42]) whereg3(βk) = g3(αk −αk+1) = expcos2π3 βk + φ/T(7.4)and φ is a fixed phase. We note the identitiesg2(0)g2(1) = 1 , g3(0)g3(1)g3(2) = 1(7.5)If the steady state is given by Eq.

(7.2), the master equation describes the evolution ofthe one-dimensional classical chain from P({β}; t = 0) = P0({β}) to the equilibriumprobability distribution P({β}; t = ∞) = P({β}).In order to find the conditions on the states, we first make a similarity transfor-mation Eq. (2.7), takingh(k)(α)=f 1/2(α)Wℓ,m(α, β)=vuutf(α + ℓ)f(β + m)f(α)f(β)wℓ,m(α, β)(7.6)32

Then the condition to have Eq (7.1) readsw0,0(α, β) =N−1X′ℓ,m=0wℓ,m(α, β)f(α + ℓ)f(β + m)f(α)f(β)(7.7)where the prime indicates that the case ℓ= m = 0 should be excluded. Taking intoaccount the probability conservation condition Eq.

(2.4) and making the change offunctionswℓ,m(α, β) =γℓ,m(α, β)f(α + ℓ)f(β + m)(7.8)we get the system of equationsN−1X′ℓ,m=0(γℓ,m(α, β) −γℓ,m(α −ℓ, β −m)) = 0(7.9)We note that in Eq. (7.9) there is no longer any explicit reference to the function f.The whole f-dependence is in Eq.

(7.8).We now write down the independent conditions. To do so, it is convenient tointroduce the following symmetric and antisymmetric combinationsγsℓ,m(α, β)=γℓ,m(α, β) + γm,ℓ(β, α)γaℓ,m(α, β)=γℓ,m(α, β) −γm,ℓ(β, α)(7.10)Now, for the two-state system the independent conditions coming from Eq.

(7.9) areγs1,0(0, 0) −γs1,0(1, 0) = γs1,0(0, 1) −γs1,0(1, 1) = γ1,1(1, 1) −γ1,1(0, 0)(7.11)γa1,0(1, 0) + γa1,0(1, 1) + γa0,1(1, 0) + γa0,1(0, 0) = 2γa1,1(0, 1)(7.12)Note that Eq. (7.11), which relates only symmetric combinations, does not, as ex-pected, contain the diffusion constant since it fixes the time scale only.

The antisym-metric diffusion combinations appear however in Eq. (7.12) and thus play a dynamicalrole.

We remind the reader that this property was already noticed in earlier sectionswhen discussing Hamiltonians with quantum group symmetries (q ̸= 1).For the three-state system with Z3 symmetry (which means that only the functionsw1,2 and w2,1 appear) we have from Eq. (7.9) the following conditionsγs2,1(0, 0) = γs2,1(1, 2) , γs2,1(1, 1) = γs2,1(2, 0) , γs2,1(2, 2) = γs2,1(0, 1)(7.13)2γa2,1(1, 0)=γa2,1(2, 2) + γa2,1(0, 1)2γa2,1(0, 2)=γa2,1(1, 1) + γa2,1(2, 0)2γa2,1(2, 1)=γa2,1(0, 0) + γa2,1(1, 2)(7.14)We observe that the detailed balance equations do not determine all the rates.

Theremaining freedom can be used to fit the experimental data. We shall apply these33

conditions on the rates in the next two sections. Before doing so, we briefly recall thequestions of interest and some definitions relevant to critical dynamics.

As alreadymentioned in the introduction, a classical Ising spin system has no intrinsic dynamics,because the Poisson brackets between the spin variables and the Hamiltonian vanish.In a real magnetic system, the spins are interacting with other degrees of freedom(phonons, impurities, . .

. ) and those interactions are responsible for the dynamics ofthe spins.

It is very difficult to implement fully the microscopic dynamics. Accord-ingly, one replaces the true dynamics by a fictious one given by a master equationfor the probability that one spin configuration of the system is realized at a giventime.

The transition probabilities appearing in the master equation can in principlebe computed from the full microscopic dynamics (via projective techniques). Un-fortunately, it is often not possible to perform this program, even for simple models[43, 44].

Thus, the transition probabilities are chosen phenomenologically accordingto two criteria: i) to satisfy detailed balance, necessary to ensure that the desiredequilibrium state is stationary, ii) to be qualitatively in agreement with the physicsof the system. Once the dynamics is defined, the main problem is to find how thephysical quantities will relax towards equilibrium.

A quantity of particular interestis the order parameter (the magnetization for the Ising model). Typically, the orderparameter relaxes very slowly in the vicinity of a second-order phase transition (criti-cal slowing-down).

The dynamical scaling hypothesis assumes that the characteristictime τ diverges as a power law of the spatial correlation length ξ; τ ∼ξz, where z iscalled the dynamical exponent.The quantum chain formalism developed here aims to answer two types of ques-tions:1. Does the dynamical scaling hypothesis τ ∼ξz always hold ?2.

What is the status of the “universality” for the dynamical exponent z ?For one-dimensional systems, the critical point is at zero temperature, Tc = 0.For small T, the inverse spatial correlation lengths areξ−1=2e−2/T ,two-state systemξ−1=32 g3(1)g3(0) + g3(2)g3(0)!,three-state system(7.15)For the three-state system we shall only consider the ordinary Potts model, whereφ = 0 and the so-called superintegrable model [45], where φ = π/6. This serves toillustrate how the dynamics depends on the equilibrium system.

In the sequel, thedeviation from the critical point will be parametrized in terms of the following massesµ ∼ξ−1, which are related to the temperature as followsµ = e−2/TIsing modelµ = e−3/(2T)Potts model, φ = 0µ = e−√3/(2T)superintegrable model, φ = π/6(7.16)34

for the three equilibrium models under consideration. In the next two sections weshall concentrate on the behaviour for small values of µ in the master equation andin the corresponding Hamiltonians.8Critical dynamics for the Ising modelWe will write the Hamiltonian corresponding to the equilibrium distribution Eq.

(7.3)and the conditions Eqs. (7.11, 7.12) for the rates.We assume that the left-rightantisymmetric combinations are zero such that Eq.

(7.12) is trivially satisfied. Tosimplify the problem further, we choose the following solution of Eq.

(7.11)γs1,0(0, 0) −γs1,0(1, 0) = γs1,0(0, 1) −γs1,0(1, 1) = γ1,1(1, 1) −γ1,1(0, 0) = 0(8.1)With this choice we getw1,0(1, 0) = µw1,0(0, 0) , w1,0(1, 1) = µw1,0(0, 1) , w1,1(1, 1) = µ2w1,1(0, 0)(8.2)This tell us that at T = 0(µ = 0) the master equation describing the critical dynamicsof the Ising model reduces to the master equation considered in Sec. 4 in whichannihilation, coagulation and death processes were considered.

The particular casewith only annihilation was already noticed by Family aand Amar [32].Taking take w1,1(1, 0) = 1 and by using probability conservation we getw0,0(0, 0)=µ2w1,1(0, 0) + 2µw1,0(0, 0)w0,0(1, 0)=1 + w1,0(0, 0) + µw1,0(0, 1)w0,0(1, 1)=2w1,0(0, 1) + w1,1(0, 0)(8.3)Using Eqs. (4.47), (4.48) and (8.2) we get the Hamiltonian:H=−12L−1Xi=1hσxi σxi+1 + σyi σyi+1 + ∆σzi σzi+1 + (1 −∆′)σzi + σzi+1+µw1,1(0, 0)σxi σxi+1 −σyi σyi+1+2µ1/2w1,0(0, 0)σxi E00i+1 + E00i σxi+1+2µ1/2w1,0(0, 1)σxi E11i+1 + E11i σxi+1i(8.4)where∆=1 −1 + µ22w1,1(0, 0) −(1 −µ)w1,0(0, 1) + (1 −µ)w1,0(0, 0)∆′=∆−µw1,0(0, 1) + µ2w1,1(0, 0) −(1 −2µ)w1,0(0, 0)(8.5)This result is very interesting and deserves a few comments.

If w1,0(0, 0) ̸= 0, evenfor µ = 0 (when the equilibrium system is at the critical temperature T = 0) we35

still have ∆′ < ∆and consequently the time-dependent system is massive with anexponential fall-offin the correlation functions. This is to the best of our knowledgethe first example of a system having an equilibrium second order phase transitionbut no critical slowing down.

It is not obvious that this phenomenon generalizes intohigher dimensions since only in one dimension, at T = Tc the system is fully ordered.On the other hand if w1,0(0, 0) = 0, but w1,0(0, 1) is non-zero, we would expectthe relaxation time should beτ −1 ∼µ(8.6)The reason is simple and can be understood using perturbative arguments. We getterms of order O(µ) in ∆′, which couples to the U(1) scalars, σzi + σzi+1, in theHamiltonian.

The terms present in the Hamiltonian which are of order O(µ1/2) occurin combinations like µ1/2w1,0(0, 1) which do not couple to U(1) scalars and shouldthus only contribute in second order. The contribution of order O(µ) can be onlyeliminated when a Z2 symmetry is present in the problem.

This implies thatw1,0(0, 0) = w1,0(0, 1) = 0(8.7)and by repeating the same argument we findτ −1 ∼µ2(8.8)However these perturbative arguments should be considered with some care sincethey apply to chains of a given length L. It might happen that in the large L limit,the leading term in the power expansions in µ has vanishing coefficients and we haveto consider the next leading order term. To be more explicit, in a concrete calculationof the case w1,0(0, 1) ̸= 0, w1,0(0, 0) = w1,1(0, 0) = 0 for a given chain one obtains:τ −1L= aL + bLµ + cLµ2 + ...(8.9)In the large L limit aL must vanish (we end up in a massless system at µ = 0 ) butthe same can also happen to bL.

It turns out that this is indeed happening sincewe can derive the exact result using the calculations of Ref. [30] for the coagulation-decoagulation chemical process and findτ −1 = µ24 .

(8.10)We would like to emphasize that the calculation of the coefficients aL, bL, cL, ... canbe difficult in general so that numerical estimates might be useful here. For the casew1,1(0, 0) ̸= 0 and w1,0(0, 1) = w1,0(0, 0) the perturbative argument is correct andEq.

(8.8) stays valid as we can see from the exact result of Ref.[31]. We conclude thatin the Ising case the critical exponent z = 2 is universal.36

9Three-states critical dynamicsTurning back to the three-states models, we now assume, in analogy to the two-statecase, that the assymmetric rates in Eq. (7.14) are zero and we thus only have to solveEq.

(7.13). We begin by considering the ordinary Potts model, with φ = 0.

We findw2,1(1, 2) = µ2w2,1(0, 0) , w2,1(1, 1) = µw2,1(2, 0) , w2,1(2, 2) = µw2,1(0, 1)(9.1)This leads to the following HamiltonianH=L−1Xi=1h2µ2w2,1(0, 0)E00i E00i+1 + 2w2,1(2, 0)E11i E11i+1 + 2w2,1(0, 1)E22i E22i+1+ (w2,1(1, 0) + µw2,1(0, 1))E00i E11i+1 + E11i E00i+1+ (w2,1(0, 2) + µw2,1(2, 0))E00i E22i+1 + E22i E00i+1+ (w2,1(2, 1) + w2,1(0, 0))E11i E22i+1 + E22i E11i+1−µw2,1(0, 0)E10i E20i+1 + E01i E02i+1 + E20i E10i+1 + E02i E01i+1−µ1/2w2,1(2, 0)E10i E12i+1 + E01i E21i+1 + E21i E01i+1 + E12i E10i+1−µ1/2w2,1(0, 1)E02i E12i+1 + E20i E21i+1 + E21i E20i+1 + E12i E02i+1−w2,1(1, 0)E10i E01i+1 + E01i E10i+1−w2,1(2, 1)E21i E12i+1 + E12i E21i+1−w2,1(0, 2)E02i E20i+1 + E20i E02i+1i(9.2)In writing Eq. (9.2), we have used the similarity transformation (2.8) with h(k)(1)/h(k)(0) =h(k)(2)/h(k)(0) = √µ.

We are thus left, after choosing the time scale, with five freeparameters. If we now take µ = 0 in Eq.

(9.2), we get back to the chains studiedin Section 6. If we make the choice of coupling constants from Eqs.

(6.10, 6.12), atµ = 0 we find that H coincides with the H0 of Eq. (6.13).

We can thus writeH = H0 + ˜H1(9.3)where ˜H1 is in the case αAA = αBB˜H1=L−1Xi=1h2µ2αABE00i E00i+1 + µαAA(1 −ρ0)iρ0i+1 + ρ0i (1 −ρ0)i+1−µαABE10i E20i+1 + E01i E02i+1 + E20i E10i+1 + E02i E01i+1−µ1/2αAAE10 + E02i E12i+1 + E12iE10 + E02i+1+E01 + E20i E21i+1 + E21iE01 + E20i+1(9.4)37

andρ0 =100000001= E00 + E22(9.5)Notice that the terms which couple to µαAA keep the U(1) ⊗U(1) symmetry of theunperturbed problem. If this is the only perturbation present, we should thus getτ −1 ∼µ(9.6)This result is similar to the behaviour observed for the Ising model (see Eq.

(8.6)).We would like to stress that this result might be ”naive” (see the discussion afterEq. (8.8) in Sec.

8).We now consider the superintegrable model, where φ = π/6. This case is interest-ing because g3(2) = 1 and this will have important consequences.

From Eq. (7.13),we findw2,1(1, 2) = µ3w2,1(0, 0) , w2,1(1, 1) = µ3w2,1(2, 0) , w2,1(2, 2) = w2,1(0, 1)(9.7)The Hamiltonian isH=L−1Xi=1h2µ3w2,1(0, 0)E00i E00i+1 + 2w2,1(2, 0)E11i E11i+1 + 2w2,1(0, 1)E22i E22i+1+ (w2,1(1, 0) + w2,1(0, 1))E00i E11i+1 + E11i E00i+1+w2,1(0, 2) + µ3w2,1(2, 0) E00i E22i+1 + E22i E00i+1+ (w2,1(2, 1) + w2,1(0, 0))E11i E22i+1 + E22i E11i+1−µ3/2w2,1(0, 0)E10i E20i+1 + E20i E10i+1 + E01i E02i+1 + E02i E01i+1−µ3/2w2,1(2, 0)E10i E12i+1 + E01i E21i+1 + E21i E01i+1 + E12i E10i+1−w2,1(0, 1)E02i E12i+1 + E20i E21i+1 + E21i E20i+1 + E12i E02i+1−w2,1(1, 0)E10i E01i+1 + E01i E10i+1−w2,1(2, 1)E21i E12i+1 + E12i E21i+1−w2,1(0, 2)E02i E20i+1 + E20i E02i+1i(9.8)Notice that for µ = 0 we got a new term in Eq.

(9.8) which did not exist in theHamiltonians studied in Section 6 and which readsw2,1(0, 1)ρ+i σ+i+1 + ρ−i σ−i+1 + σ−i ρ−i+1 + σ+i ρ+i+1(9.9)where we have used the notation from Eq. (6.20).It is not known whether theHamiltonian Eq.

(9.8) for µ = 0 is critical or not. Comparing the ordinary Pottsand the superintegrable Potts cases we notice that the number of independent rates38

is different. We also see, by comparing Eqs.

(9.2) with (9.9) that µ1/2 is replacedby µ3/2, consequently the dynamical exponents changes.We conclude that whenmoving from the Potts model (symmetry S3) to the superintegrable chiral Pottsmodel (symmetry Z3), the exponents also changes.We have seen how the universal critical exponents can be predicted from con-sidering the structure of integrable Hamiltonians at µ = 0 and appealling to per-turbation theory.Here again, we have implicitly assumed that all the six ratesw2,1(0, 0), w2,1(2, 0), w2,1(0, 1), w2,1(1, 0), w2,1(2, 1) and w2,1(0, 2) have finite, non-va-nishing limits as µ →0.Otherwise, one may create any value of some effectiveexponent like in Refs. [18, 43, 44].10ConclusionsWe have started our study asking ourselves what we thought is a simple question:is the present progress achieved in the understanding of one-dimensional integrablequantum chains useful for solving master equations describing the dynamics of classi-cal one-dimensional spin systems ?

We find plenty of evidence for a positive answer.At same time, after finishing this long paper, we have the feeling that we are just atthe beginning of a long path.Although initially we thought that our task would be just to use the availablemathematical knowledge of integrable systems to find physical results, the physicalproblems brought a lot of feed-back into mathematics. Studying open chains withparticular transitions rates lead us, to our surprise, automatically to new, hermitianand non-hermitian, representations of interesting associative algebras.

We remindthe reader that in equlibrium problems, quantum groups and associative algebrasappear only through rather artificial boundary conditions. The physical applicationsof non-hermitian representations of associative algebras appear for the first time inour context.

More about this subject will be published elsewhere.We would like to stress once more that although the phase diagram in the spaceof transition rates can be easily obtained from the Hamiltonian spectrum computedfor example by using the Bethe ansatz, the calculation of nonequilibrium averages(which are not those normally occuring in equilibrium statistical physics) might poseformidable problems.For the physical understanding of reaction-diffusion processes or of simple criticaldynamics we think that we went beyond the particular cases studied up to now inthe literature.As the reader has certainly noticed while going through Sections 4 and 6, there isplenty of room for more work especially for three-states models, where we have onlyconsidered systems with Z3 symmetry and even in this case, the whole phase diagramis not yet completely known.Our experience with the study of properties of quantum chains for equlibriumpurposes lead us to repeat analogous questions for nonequlibrium problems. For ex-39

ample, in the simple model studied in Section 5 we have verified that finite-size scalingapplies also to nonequlibrium situations. This opens the possibility of using standardnumerical methods of matrix diagonalization of finite systems, for computing criticalexponents.

The solutions found in Section 7 for the detailed-balance equations shouldbe useful for other dynamical processes than those discussed in Sections 8 and 9, ina better understanding of critical dynamics.AcknowledgementsWe would like to thank C. G´omez, V. Privman and G. Sch¨utz for useful discus-sions. We are most grateful to Klaus Krebs, Markus Pfannm¨ulller and Birgit We-hefritz for the excellent job they did critically reading the manuscript.

F.A. thanksthe Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo - FAPESP - Brasil andthe Deutsche Forschungsgemeinschaft - DFG - Germany for support.

M.D. and M.H.thank the Swiss National Science Foundation for support.

V.R. would like to thankCERN, the University of Geneva, the Einstein Center of the Weizmann Institute andSISSA for the warm hospitality he has enjoyed and has enabled him to participate inthis project.Appendix A. Eigenvectors of some special non-hermitianHamiltoniansWe discuss the treatment of a certain class of non-hermitian Hamiltonians with thestructureH = H0 + H1(A.1)where H0 is hermitian and has the same spectrum as H. Moreover we are interestedin the case where the eigenspectrum is non-degenerated.

Suppose the eigenvectors|ui⟩and the eigenvalues λi of H0H0 |ui⟩= λi |ui⟩(A.2)are known and we want to find the eigenvectors of H. This problem is of interest forboth analytical and numerical (finite-size scaling) calculations in reaction-diffusionprocesses (see Eqs. (4.11), (4.31) and (6.11)).

In the basis {|ui⟩}, H1 has only thefollowing non-vanishing matrix elements⟨ui| H1 |uj⟩= Gij ; i < j , i, j = 1, . .

., N(A.3)40

Thus the eigenvalues of H are again λ1, . .

., λN. Let |vi⟩be the eigenvector of Hcorresponding to the eigenvalue λi.

It can be written as|vi⟩=Xi≥jAij |uj⟩(A.4)with Aii = 1 and the graphical rule to compute the Ai,j(i > j) is obvious if we givethe first fewA21=G12λ2 −λ1, A32 =G23λ3 −λ2, A31 =G13λ3 −λ1+G12G23(λ3 −λ2)(λ3 −λ1)A43=G34λ4 −λ3, A42 =G24λ4 −λ2+G23G34(λ4 −λ3)(λ4 −λ2)A41=G14λ4 −λ1+G13G34(λ4 −λ3)(λ4 −λ1) +G12G24(λ4 −λ2)(λ4 −λ1)+G12G23G34(λ4 −λ3)(λ4 −λ2)(λ4 −λ1)(A.5)Appendix B. The seven-vertex modelAs an example, we shall consider in more detail one of the models related to a Heckealgebra (see Appendix C) in order to obtain the Boltzmann weights of an associatedtwo-dimensional vertex model and thus prove its integrability.The model we consider is the one introduced in Section 4 which describes diffusionand pairwise annihilation of A particles, see Eq.

(4.8). We choose the special tuningwhich makes ∆= 0 and take the functions ϕ(k)(α, β) = 1 where α, β = 0, 1.

TheHamiltonian is given byH = D(H0 + H1)(B.1)whereH0=−L−1Xi=1"qE01i E10i+1 + 1qE10i E01i+1 + q2 (σzi −1) + 12qσzi+1 −1#H1=−ΩL−1Xi=1E01i E01i+1(B.2)andΩ= w1,1(0, 0)D, q =vuutw1,1(1, 0)w1,1(0, 1) , D =qw1,1(0, 1)w1,1(1, 0)(B.3)Doing the canonical transformation Ekℓi= (−1)k−ℓEkℓi , only at even sites i, thisHamiltonian in the σz basis takes the simple formH = −DL−1Xi=1ei(B.4)41

whereei = 11 ⊗. .

. ⊗1i−1 ⊗000Ω01qq001qq0000q + 1q⊗1i+2 ⊗.

. .

(B.5)and i = 1, 2, . .

., L −1 and 1i are 2 × 2 unit matrices attached to the site i.The Hamiltonian Eq. (B.4) is known to be integrable through a Jordan-Wignertransformation (see Section 5), we think however that the approach given here givesnot only new insight into the problem but is of a larger validity.We can show that the above matrices ei, i = 1, .

. .

, L satisfy the Hecke algebra,for arbitrary values of Ωeiei±1ei −ei = ei±1eiei±1 −ei±1[ei, ej] = 0 ; |i −j| ≥2(B.6)e2i =q + q−1eiThis is the first example we know of where nonhermitian (take q real) representationsof the Hecke algebra appear in physical applications.The hermitian case Ω= 0corresponds to the quantum chain introduced by Saleur [46].Due to the algebraic properties Eq. (B.6) we can construct an associated two-dimensional vertex model having a row-to-row transfer matrix depending on thespectral parameter θ.

There transfer matrices will satisfy the Yang-Baxter equations[47] which implies that they commute among themselves for different values of thespectral parameter.In order to obtain the configuration and the Boltzmann weights associated to thisvertex model, we need the spectral parameter dependent matrix ˇRi(θ), i = 1, 2, . .

..This is found by the Baxterization procedure [21] for Hecke algebras, namelyˇRi(θ) = sinh θsinh ηei + sinh(η −θ)sinh η, q = eη(B.7)which gives usˇRi(θ) = 11⊗. .

.⊗1i−1⊗1sinh ηsinh(η −θ)Ωsinh θe−θ sinh ηeη sinh θe−η sinh θeθ sinh ηsinh(η + θ)⊗1i+2⊗. .

. (B.8)The relations Eq.

(B.6) imply that these matrices satisfy the spectral parameterdependent braid group relationsˇRi(θ) ˇRi±1(θ + θ′) ˇRi(θ′)=ˇRi±1(θ′) ˇRi(θ + θ′) ˇRi±1(θ)h ˇRi(θ), ˇRj(θ′)i=0 , |i −j| ≥2(B.9)42

which are equivalent to the Yang-Baxter equations.The Boltzmann weights Sknℓ,m of the vertex configuration labelled by (k, ℓ, m, n) inthe associated vertex model can be obtained from the relationˇRi(θ) = Sk,nℓ,m11 ⊗. .

. ⊗1i−1 ⊗Emk ⊗Enℓ⊗1i+2 ⊗.

. .

(B.10)This implies that the vertex model associated to Eq. (B.1) is a seven vertex model.If we denote by an index zero a down (or left) arrow and by an index one an up(or right) arrow, the vertex configurations with their Boltzmann weights are given inFigure 2.

We also show in this figure the corresponding chemical processes relatedto each vertex. The vertices 1 to 4 correspond to no reaction, vertices 5 and 6 todiffusion to the right and the left and vertex 7 to the pair annihilation process.

Thederivative of the logarithm of the row-to-row transfer matrix, with these Boltzmannweights and evaluated at θ = 0, gives back the Hamiltonian Eq. (B.1).Appendix C. Hecke algebra and reaction-diffusionprocessesIn this appendix we define the Hecke algebra and give some examples of Hamiltonians,related to dynamical processes, which are representations of this algebra.The Hecke algebra is an associative algebra with generators ei (i = 1, .

. ., L −1)satisfying the relationseiei±1ei −ei = ei±1eiei±1 −ei±1(C.1)[ei, ej] = 0 ; |i −j| ≥2(C.2)e2i =q + q−1ei(C.3)where q is a complex parameter.One of the main features of the above algebra is related to the fact that a spectral-dependent ˇR(u) matrix, satisfying the Yang-Baxter relations [47], can be constructedin a standard form.

As a consequence of this procedure, also called “Baxterization”[21], the Hamiltonian PL−1i=1 ei, as well as its associated vertex model, has an infinitenumber of conservation laws and we expect, in general, its exact integrability. InAppendix B we give an example of this “Baxterization” procedure and derive theassociated vertex model for one of the chains considered in this paper.It is important to stress here that distinct Hamiltonians satisfying the same Heckealgebra may correspond to different representations of the algebra.

They will share,apart from degeneracies (which may be zero) the same eigenenergies. These chainswill have a massive or massless behaviour depending on the value of q.

In particular,for q real they will always have a massive behaviour except for q = 1, where they willbe massless.The ei’s appearing in quantum chains quantum chains satisfy beyond Eqs. (C.1-C.3) additional relations which define quotients of the Hecke algebra.Obviously,43

quantum chains obeying the same quotient have more coincidences in their spectra,see [23] for details. A well-known quotient is the Temperley-Lieb algebra [48] definedby (C.2,C.3) andeiei±1ei = ei , ei±1eiei±1 = ei±1 ; i = 1, 2, .

. .

(C.4)A realization of this algebra is given by the XXZ chain with surface magnetic fields(invariant under the UqSU(2/0) quantum group) and the quantum Potts chain withfree ends [48, 49].In Sections 4 and 6 we have used the notation UqSU(n) forUqSU(n/0). With the new notation UqSU(n/m) with m ̸= 0 corresponds to quantumsuperalgebras [50].) Another less known quotient is defined by (C.1-C.3) and theadditional relation [23](eiei+2) ei+1q + q−1 −ei q + q−1 −ei+2= 0 ; i = 1, 2, .

. .

(C.5)which has a realization in the two-colors Perk-Schultz model [22], invariant under thequantum group UqSU(1/1).In this paper we show that several quantum chains, related to chemical processes,are realizations of the Hecke algebra (C.1)-(C.3). As a general outcome from ouranalysis we verifyed that as long as only diffusion and interchange of particle processes(the number of particles in each species is conserved separately) are allowed, theserealizations arise quite naturally.

In the cases where other processes are also allowedthe chain in general will not satisfy the Hecke algebra. However, as we shall see,for certain processes and special tunings of transition rates, the corresponding chainswill turn out to satisfy the Hecke algebra again.Although we can generalize our results to an arbitrary number of different chemicalspecies, following the line of this paper we will consider here only the cases where wehave, beyond vacancies, 1 species (A) or 2 species (A and B).Let us consider initially the cases where we have only particles and vacancies, seeSection 4.1.

If we only allow the diffusion process with the transition rates satisfyingEqs. (4.9), (4.11) we obtain the XXZ chain with anisotropy ∆= (q + q−1)/2H/D = H0 =LXi=1ei(C.6)whereei=−12"σxi σxi+1 + σyi σyi+1 + 12 q + 1q!σzi σzi+1 + 12 q −1q!

σzi −σzi+1−12 q + 1q!#; i = 1, . .

. , L −1(C.7)We can verify that these ei matrices are just a 2L-dimensional representation ofthe Hecke algebra.

Moreover, this Hamiltonian satisfies the quotient defining the44

Temperley-Lieb algebra (C.4). It is important to stress that the terms proportional toσzi appear naturally in (C.7) due to the diffusion mechanisms.

These terms, althoughnot present in a periodic chain, are crucial in order to generate the Temperley-Liebalgebra. The particular chain (C.7) is invariant under the quantum group UqSU(2/0).Another example, also with only particles and vacancies, appears when beyonddiffusion we also allow the annihilation process A + A →∅+ ∅(with rate w1,1(0, 0)).In this case, by making the special tuningw1,1(0, 0) = w1,1(0, 1) + w1,1(1, 0)(C.8)and choosing the functions ϕ(k)(ℓ, m) = 1 we obtainH/D =L−1Xi=1ei(C.9)whereei=−qE01i E10i+1 −1qE10i E01i+1 − q + 1q!E01i E01i+1+q2 (σzi −1) + 12qσzi+1 −1; i = 1, .

. .

, L −1(C.10)andq =vuutw1,1(1, 0)w1,1(0, 1) , D =qw1,1(0, 1)w1,1(1, 0)(C.11)These matrices ei satisfy the Hecke relations (C.1)-(C.3) and are also generators of thequotient defined by Eq. (C.5).

This is the first example we are aware of in which non-hermitian (q real) representations of the Hecke algebra appear in a physical context.In Appendix B we derive the vertex model associated to this chain.Let us now consider some cases where we have two types of particles (A andB). If we consider only the process of diffusion and interchange of particles, as wesaw in Section 6, by choosing the diffusion rates Eq.

(6.4) we obtain the anisotropicSutherland model [36]H/D =L−1Xi=1ei(C.12)whereei=12 q + 1q!−2Xa,b=0;a̸=bEabi Ebai+1 + 12 q + 1q!2Xa=0Eaai Eaai+1+12 q −1q!2Xa,b=0;a̸=bsign(a −b)Eaai Ebbi+1(C.13)which again satisfy the Hecke algebra. The above Hamiltonian is invariant underthe quantum group UqSU(3/0) (we do not give the corresponding quotient here).45

Let us now consider the case where beyond the above processes we also include theannihilation A + A →∅+ ∅(rate w1,1(0, 0)). If we use the relation (6.5) between thediffusion rates and the conditionw1,1(0, 0) = ΓL + ΓR(C.14)we obtainH/D′ =L−1Xi=1ei(C.15)whereei=−"1qE01i E10i+1 + qE10i E01i+1 + 1qE02i E20i+1+qE20i E02i+1 + 1qE12i E21i+1 + qE21i E12i+1 + ΩE01i E01i+1#+qE00i E11i+1 + E00i E22i+1 + E11i E22i+1+ 1qE11i E00i+1 + E22i E00i+1 + E22i E11i+1+ q + 1q!E11i E11i+1(C.16)andΩ= q + 1q , q =qΓR/ΓL , D′ =qΓRΓL(C.17)The (nonhermitian!) operators ei also satisfy the Hecke algebra.

In fact we verifiedthat this is a property of (C.16)-(C.17) for arbitrary values of Ω. For Ω= 0 theHamiltonian reduces to the three-color Perk-Schultz model [22] (which has the quan-tum superalgebra UqSU(2/1) as symmetry) and it is also related to a special pointof the t −J model [51] where exact integrability takes place [52].Let us return to the case where we do not have annihilation, see Eqs.

(C.9,C.11). Ifwe now forbid the process where the particles interchange positions (A+B ↔B +A)w1,2(1, 2) = w2,1(2, 1) = 0(C.18)we obtainH/D′ =L−1Xi=1ei(C.19)withei=−" 2Xa=1E0ai Ea0i+1 + Ea0i E0ai+1+ qEi00 E11i+1 + E22i+1−1qE11i + E22iE00i+1i; i = 1, .

. .

, L −1(C.20)46

This model was introduced in [53] and the generators satisfy the Hecke algebra.Moreover we verified that like the UqSU(2/0) model (C.6,C.7) the matrices ei arealso the generators of a 3L-dimensional Temperley-Lieb algebra. The symmetry ofthe chain (C.20) is known and described in Ref.

[55].If we now include (see Eq. (6.11)) the annihilation processes A + B →∅+ ∅,A + A →∅+ B, B + B →∅+ A with the reaction rates related in the special wayw2,1(1, 0) = w1,2(2, 0) = w2,1(2, 0) = w1,2(1, 0) = ΓL(C.21)w1,2(0, 1) = w2,1(0, 2) = w1,2(0, 2) = w2,1(0, 1) = ΓR(C.22)w1,2(0, 0) = w2,1(0, 0) = ΓR + ΓL(C.23)q =qΓL/ΓR , D′ =qΓLΓR(C.24)we obtainH/D′ = H0/D′ + H1/D′ =L−1Xi=1ei , ei = e0i + e1i(C.25)wheree0i=2Xa=1 qE00i Eaai+1 + 1qEaai E00i+1 −qEa0i E0ai+1 −1qE0ai Ea0i+1!+ q + 1q!2Xa,b=1Eaai Ebbi+1 ; i = 1, .

. ., L −1(C.26)ande1i=− 1qE01i E21i+1 + qE21i E01i+1 + qE12i E02i+1 + 1qE02i E12i+1+ q + 1q!E01i E02i+1 + q + 1q!E02i E01i+1!

(C.27)The Hamiltonian H0/D′ = PL−1i=1 e0i was introduced in [53], (see also Ref. [54] for thecase q = 1) as a 3L-dimensional representation of the Hecke algebra.

We verified,guided by the physical processes of diffusion and annihilation, the the operators ei(i = 1, . .

., L −1) also satisfy the same algebra. Morevover, we checked that bothe0i and ei satisfy the quotient (C.5) like the UqSU(1/1) chains, for the underlyingquantum symmetry of (C.25) see Ref.

[55].Figure captionsFigure 1: Phase diagram of the XXZ quantum chain with anisotropy ∆in a magneticfield h. I is the massive (frozen) ferromagnetic phase, II the massless phase which47

is commensurate for h = 0 and incommensurate for h ̸= 0 and III is the massive(frozen) antiferromagntic phase. The line ∆= 1 −h corresponds to a Pokrovsky-Talapov transition.Figure 2: Boltzmann weights of the two-dimensional vertex model associated to theHamiltonian Eq.

(B.1). The first column gives the weights, the second one the arrowdiagram and the third one the relationship between the vertices and the elementaryreaction-diffusion processes.References[1] A. Tsuchiya, T. Eguchi and M. Jimbo (Eds) ”Infinite analysis”, Part A and B,World-Scientific,Singapore, 1992.

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