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Critical Theories of the Dissipative Hofstadter Model

arXiv:hep-th/9202085v1 24 Feb 1992PUPT-1292CTP #2073hepth@xxx/9202085Critical Theories of the Dissipative Hofstadter ModelCurtis G. Callan†,Andrew G. Felce†Department of Physics, Princeton UniversityPrinceton, NJ 08544Denise E. Freed††Center for Theoretical Physics, MITCambridge, MA 02139AbstractIt has recently been shown that the dissipative Hofstadter model (dissipative quantummechanics of an electron subject to uniform magnetic field and periodic potential in twodimensions) exhibits critical behavior on a network of lines in the dissipation/magneticfield plane. Apart from their obvious condensed matter interest, the corresponding criticaltheories represent non-trivial solutions of open string field theory, and a detailed accountof their properties would be interesting from several points of view.

A subject of particularinterest is the dependence of physical quantities on the magnetic field since it, much likeθQCD, serves only to give relative phases to different sectors of the partition sum. In thispaper we report the results of an initial investigation of the free energy, N-point functionsand boundary state of this type of critical theory.

Although our primary goal is the studyof the magnetic field dependence of these quantities, we will present some new resultswhich bear on the zero magnetic field case as well.2/92† callan@puhep1.princeton.edu, felce@puhep1.princeton.edu†† freed@mitlns.mit.edu

1. IntroductionThe Hofstadter problem concerns the quantum mechanics of an electron moving intwo dimensions subject to a magnetic field and a periodic potential.

The energy bandsof this model show a remarkable fractal structure [1,2,3] as a function of the number offlux quanta per lattice unit cell.In a previous paper [4], the Caldeira-Leggett model[5] of dissipative quantum mechanics (DQM) was used to study how this discontinuousbehavior is smoothed out by the unavoidable elements of randomness in real physicalsystems. A complicated phase diagram was discovered which showed precisely how this“smoothing out” works: Above a certain critical dissipation, the particle is localized, but,as the dissipation is reduced, there is an increasingly dense system of phase-transitionlines which have a fractal structure in the zero-dissipation limit.

The topic of this paperis the study of the properties of the one-dimensional critical theories corresponding to thephase transitions themselves. In addition to their relevance to the Hofstadter problem,these theories represent new solutions of open string theory in a non-trivial backgroundof tachyons and gauge fields [6].

The string theory connection strongly suggests that thecritical theory should have an enhanced SL(2, R) invariance in addition to the usual scaleinvariance and we will verify that this is so.We have found a simple regulator which reduces the calculation of most quantities ofinterest, to any order of perturbation in the potential, to the purely algebraic exercise ofextracting the residues of poles in a rational function associated with each perturbationtheory diagram. Using this regulator, for some special points in the phase diagram weexplicitly demonstrate the absence of logarithmic divergences, and show that the onlyrenormalization needed is a rescaling of the potential strength and the subtraction of aninfinite constant from the free energy.

We give evidence for the criticality of the circulararcs in the phase diagram which join these special points to one another, although wecannot give a proof to all orders in this case. In addition we have done some explicitcalculations of the magnetic field dependence of free energies and N-point functions tovarious orders in the potential strength.

We find that many of the connected higher N-point functions are zero, up to contact terms. Although this suggests that the criticaltheories are “almost” free and therefore ought to be soluble, we have not been able toexploit this hint to obtain exact solutions and must for the moment content ourselves withthe rather clumsy perturbative approach presented here.In the interests of making this paper relatively self-contained, we devote the firsttwo sections to a brief review of background material that has, by and large, appeared1

elsewhere. In Section 2 we review dissipative quantum mechanics and its relationship withopen string theory.

In Section 3 we specialize to the Hofstadter model. We show that it isequivalent to a Coulomb gas, demonstrate that it has phase transitions and show that thecritical theories have SL(2, R) invariance.

We then move on to the considerations which arenew to this paper. In Section 4 we give a fairly complete discussion of the unique criticaltheory at zero magnetic field, and show that it is a free theory with certain nontrivialcontact interactions.

In Section 5 we calculate the free energies and N-point functions ofcritical theories at non-zero magnetic field in the zero-charge sector of the Coulomb gas. Aninteresting feature of our results is that at the previously-mentioned special points in thephase diagram, most of the N-point functions reduce to contact terms.

There are, however,some few that do not and we give their SL(2, R)-invariant form. Section 6 is devoted tothe non-zero charge sectors of the Coulomb gas.

They are crucial to the construction ofthe open string boundary state, and we find that no new renormalizations are needed,beyond those needed to deal with the zero-charge sector. Our conclusions are in Section7.

In one appendix we present details of the proof that all the higher N-point functionsreduce to contact terms in the absence of a magnetic field. In another, we show that theN-point functions satisfy the very non-trivial string theory reparametrization invarianceWard identities.2.

Background: Dissipative Hofstadter Model and Open String TheoryWe begin with a brief outline of dissipative quantum mechanics and its connection withopen string theory. For details the reader is referred to [5] and [6].

A macroscopic objectis typically subject to dissipative forces caused by its interaction with its environment.Classically, these forces can be described by including the phenomenological term −η ˙Xiin the equation of motion for the particle, where η is the coefficient of friction and Xithe particle’s coordinate. In order to describe dissipation quantum mechanically, one canmodel the environment by a bath of an infinite number of harmonic oscillators coupledlinearly to the ⃗X.

The coupling constants, Cα, and the distribution of the frequencies,ωα, of the oscillators can be chosen so that when the oscillator coordinates are eliminatedvia their equations of motion, the resulting equations of motion for the Xi contain therequired friction term. The functional condition on the parameters isJ(ω) = πXαC2α2ωαδ(ω−ωα) = η ω ,(2.1)2

which represents Ohmic dissipation in the system. This is the Caldeira-Leggett model [5].Since the dependence on the oscillator coordinates in the lagrangian is quadratic, itis also possible to integrate them out from the quantum mechanical path integral.

Thisresults in a quantum effective action for the Xi variables, which includes a non-local piececontaining the effect of dissipation. For a particle coupled to scalar (V ) and vector ( ⃗A)potentials the full action readsS[ ⃗X] =Zdt 12M ˙⃗X2+ V ( ⃗X) + i Ai( ⃗X) ˙Xi + η4π∞Z−∞dt′ ⃗X(t)−⃗X(t′)2(t−t′)2.

(2.2)Remarkably, the only dependence on the oscillator parameters that remains in the actionis through the η-term. Because this term is non-local, the path integral is effectively thatof a one-dimensional statistical system with long-range interactions.

Such systems, unlikeone-dimensional local systems, have phase transitions (the classic example being the Isingchain with 1/r2 interactions 1). In the DQM context, the phase transitions are betweendifferent regimes of long-time behavior of Green’s functions (typically between localizedand delocalized behavior).At these critical points, the 1-D field theories describing dissipative quantum me-chanics correspond to solutions of open string theory [6].

In the presence of open stringbackground fields, interactions between a string and the background take place at theboundary of the string and their effects can be represented by a boundary state |B⟩. In[9] it is shown that this boundary state is given by|B⟩= exp( ∞Xm=11mα−m · ˜α−m) Z hD ⃗X(s)i′exp(−SR −SKE −SI −SLS)|0⟩,(2.3)whereSR =Z T0ds 12M ˙⃗X2(s) ;(2.4)SKE =18π2α′Z T0dsZ ∞−∞ds′ ⃗X(s) −⃗X(s′)2(s −s′)2;(2.5)SI = iZ T0ds Aµ( ⃗X) ˙Xµ +Z T0ds T ( ⃗X) ;(2.6)1 The Dyson chain with 1/rn interactions for n ≤2 is known to have phase transitions.

Seereferences [7,8] and references therein.3

andSLS =r2α′Zds α(s) · ⃗X(s)withαµ(s) =∞Xm=1i(˜αµ−me−ims + αµ−meims) . (2.7)In these expressions, T is the parameter length of the boundary (when appropriate, wemust regard ⃗X(s) as periodic in s with period T), α′ is the string constant and Aµ( ⃗X)and T ( ⃗X) are the gauge fields and tachyon fields, respectively.

The creation operators ofthe left- and right-moving modes of the closed string, ˜α−m and α−m, act on the closedstring vacuum |0⟩to create some state in the closed string Hilbert space. The notationD ⃗X(s)′ means that the zero-mode, ⃗X0 is not integrated out.

The commuting objects˜α−m, α−m and ⃗X0 together make up a set of coordinates which specify where the boundarylies in the target space and the boundary state is just a functional of these coordinates.As an example of the utility of this construct, we note that the projection of |B⟩ontothe graviton state is essentially the energy-momentum tensor of the open string objectunder study.This gives us a string-theoretic way to define such important notions asgravitational and inertial mass.This path integral is the generating functional for a renormalizable “one-dimensional”field theory described by the underlying action SKE +SI: (SLS is the linear source term inthe generating function and the kinetic term SR functions as a regulator for divergences).Leaving aside the linear source term, it is clear that the DQM action and the action definingthe string theory boundary state are the same if we relate the coefficient of friction to thestring tension by η = 1/(2πα′). The full string theory prescription for |B⟩requires that wetake the cut-off, M, to zero.

In order for this limit to be meaningful, the field theory mustlie at a renormalization group fixed point, which is to say that the gauge and tachyonfields must satisfy some “vanishing beta function” equations of motion for open stringbackground fields. This means that the associated DQM must lie at a phase transition.The upshot is that, modulo technical details, any solution of the open string equationsof motion is equivalent to a particular critical DQM: The background gauge and tachyonfields of the string theory become the vector and scalar potentials to which the DQMelectron is subject.In string theory we require worldsheet reparametrization invariance, and this in-cludes reparametrizations of the boundary.The condition that the boundary state bereparametrization invariant can be written(Ln −˜L−n)|B⟩= 0 .

(2.8)4

where the L operators are the closed string Virasoro generators (they act in a known wayon the coordinates α−m, ˜α−m and ⃗X0 on which the boundary path integral depends).This symmetry generates a set of Ward identities which turns out to be very useful both instring theory and in DQM (details can be found in [10]). In fact, the one-dimensional fieldtheory has what amounts to broken reparametrization invariance, because of the non-localdissipation term.

There is nonetheless a remaining manifest SL(2, R) symmetry (SU(1, 1)on the circle) which tightly constrains the allowed form of the correlation functions. Theseextra symmetries would not have been expected at the one-dimensional critical pointswithout the string theory connection and we will look for them in our explicit calculations.3.

General Properties of the Dissipative Hofstadter Model3.1. Equivalence to a Coulomb GasTo specialize to the dissipative Hofstadter model, we consider a particle moving in aperiodic potential in two dimensions (with ⃗X = (X, Y )) and subject to a uniform magneticfield B.

The Euclidean action for this problem is the sum of a quadratic piece and a morecomplicated potential term:S = Sq + SV(3.1)where1¯hSq = 1¯hZ T/2−T/2dtM2˙⃗X2+ ieB2c ( ˙XY −˙Y X) + η4π∞Z−∞dt′ ( ⃗X(t) −⃗X(t′))2(t −t′)2(3.2)andSV =Z T/2−T/2dt V (X, Y ) . (3.3)For the periodic potential we takeV (X, Y ) = −V0 cos(2πX(t)a) −V0 cos(2πY (t)a) .

(3.4)Nothing dramatically new happens if we take the strength and period of the potentialto be different in the X- and Y- directions. It is convenient to define the dimensionlessparameters2πα = ηa2¯h ,2πβ = eB¯hc a2 ,(3.5)5

to rescale X and Y by a/2π, and to rescale V0 by ¯h. Until we come to consider the infraredregulation of the theory in Section 3.4, we take T to be infinite, which means that we are atzero temperature, and that the particle lives on a line.

Then the action Sq can be writtenas1¯hSq = 12Z dω2π α2π |ω| + Ma2¯hω2δµν + β2π ǫµνω˜X∗µ(ω) ˜Xν(ω) . (3.6)Because the ordinary kinetic term, 12M ˙⃗X2, is a dimension-two operator, it is irrelevantand acts only as a regulator as far as the large-time behavior is concerned.

Since we arestudying critical behavior, it will be legitimate to set M = 0 and use some other, moreconvenient, regulator where needed. The Fourier-transformed propagator defined by (3.6)(with M = 0) isGµν(ω) = 2παα2 + β21|ω|δµν −2πβα2 + β21ω ǫµν.

(3.7)In the time domain, this becomesGµν(ti −tj) = −αα2 + β2 ln(ti −tj)2δµν −i22πβα2 + β2 sign(ti −tj)ǫµν. (3.8)For future reference, we note that in one dimension and for vanishing magnetic field thispropagator reduces toG(t1 −t2) = −1α ln(ti −tj)2.

(3.9)Except for the cosine potential, the action (3.1) is quadratic, so we will treat thepotential as a perturbation. We proceed by expandingexpV0Zcos X(t)dt=∞Xn=0Zdτ1 .

. .

dτnV02n 1n!Xqj=±1nYj=1eiqjX(τj). (3.10)Then the partition function is given byZ = exp−1¯hS=ZD ⃗X(t)∞Xn=0Zdτ1 .

. .

dτnV02n 1n!X⃗qj= (±1,0)(0,±1)nYj=1ei⃗qj· ⃗X(τj)e−Sq/¯h=∞Xn=0Zdτ1 . .

. dτnV02n 1n!X⃗qj* nYj=1ei⃗qj· ⃗X(τj)+0,(3.11)where the functional integral is over periodic paths, and the correlation functions of theoperators ei⃗q· ⃗X(t) are to be computed with the propagator (3.8).

A subtlety to be borne in6

mind is that for the dissipative quantum mechanics system, we must integrate over the zeromode in equation (3.11), which imposes the charge conservation requirement P ⃗qi = 0.For the string theory boundary state path-integral, we omit the integration over the zeromode, and the qi’s are unconstrained.If we restrict to one dimension and drop the magnetic field, the O(V0n) term in (3.11)is a sum over qj = ±1 ofZn = 1n!V02n ZnYi=1dtiDeiq1X(t1)eiq2X(t2) . .

. eiqnX(tn)E0(3.12)Using (3.9), this evaluates toZn = 1n!V02 e−12⟨X2(0)⟩n ZnYi=1dti exp 1αXi

(3.13)The expression in the exponent is the free energy for n particles with charges qj = ±1,interacting logarithmically and restricted to a line. For the dissipative quantum mechanicssystem, the condition that P qj = 0 just requires the gas to be neutral.

For the boundarystate path integral, the gas can have any charge. The full partition function describes aone-dimensional Coulomb gas of particles with fugacity proportional to V0.Similarly, in the case of most interest to us (two dimensions and non-zero field), theO(V0n) term for Z is a sum over ⃗qj = (±1, 0) and (0, ±1) ofZn = 1n!V02 e−12⟨X2(0)⟩n ZnYi=1dtiexpXi

(3.14)This has an interpretation as a generalized “Coulomb” gas. Now there are two species ofparticles, one corresponding to the X-component of ⃗q and one to the Y-component of ⃗q.Charges of the same species still interact logarithmically (through the δµν piece of (3.8)).Charges of differing species only interact through a sign function (the ǫµν piece of (3.8))and the wave-function picks up a phase factor when they are interchanged.

One of ourmain points is that this simple generalization of the Coulomb gas has a very rich phasestructure.This Coulomb gas sum has an additional interpretation. When there is no dissipation,it was shown in ref.

[11] that the partition function (3.14) describes an electron in a Landau7

orbit centered on a dual lattice site, m(a/β)ˆx+n(a/β)ˆy. Whenever the potential acts (viaan insertion of a (V0/2)ei⃗qj· ⃗X(tj)), the center of the Landau orbit hops by ⃗qj units in thereciprocal lattice and the action picks up the Aharonov-Bohm phase due to this hop.

Oncewe turn on the dissipation, according to equation (3.14), there are two major changes. Thefirst is that the dual lattice becomesmaα2 + β2 ˆx + naα2 + β2 ˆy ,(3.15)so that now, when the center of the particle’s orbit hops through a square in the reciprocallattice, it picks up a phase of ±2πβ/(α2 + β2).

Additionally, there is now a logarithmicinteraction between the particle’s hops at time ti and time tj.3.2. Phase Transitions in the Dissipative Hofstadter ModelIn standard quantum mechanics, a particle in a periodic potential is delocalized bycoherent quantum tunneling effects.

In the presence of strong enough dissipation one ex-pects coherence between tunneling events to be lost and the particle to become localized.The signature of this localization-delocalization phase transition can be looked for in theasymptotic behavior of the two-point function, ⟨X(t1)X(t2)⟩: Let the long-time behaviorof the two-point function be (t1 −t2)γ+ const. Localization corresponds to γ < 0, delo-calization to γ > 0.

When γ = 0, so that ⟨X(t1)X(t2)⟩∼ln(t1 −t2), the system is at acritical point.Fisher and Zwerger [12] have given a renormalization group argument which showsthat DQM with a periodic potential and zero magnetic field is critical at α = 1. Theirargument is valid for small values of the potential strength, V0, since they treat the potentialperturbatively, but sum over all loops.

Following this calculation closely, we can extend itto include a constant magnetic field. We will present here the calculation to first order inV0 in order to identify a candidate for a critical circle on the α-β plane.

We can show thatthe results given here remain true at order V 20 , and we expect that they will hold for allorders in V0.In the action defined in (3.1) and the following equations, we set the mass term tozero and regulate with a high frequency cut-off, Λ, instead. The action is then given by1¯hS = 12Z Λ−Λdω2π˜X†ρ(ω)Sρλ(ω) ˜Xλ(ω) + ΛV0(Λ)Zdτ [cos X(τ) + cos Y (τ)] ,(3.16)8

withSµν(ω) = α2π |ω|δµν + β2π ωǫµν(3.17)and ΛV0(Λ) replacing the bare coupling V0. At this point, we want to perform the func-tional integral over all the fast modes, ˜X(ω), with Λ > ω > µ, for some µ << Λ.

To doso, we divide the field into fast and slow modes,⃗X(τ) = ⃗Xs(τ) + ⃗Xf(τ) ,where⃗˜X(ω) ≈⃗Xs(ω)for|ω| ≤µand⃗˜X(ω) ≈⃗Xf(ω)forµ ≤|ω| ≤Λ . (3.18)We would like to calculate ˜S, where ˜S is defined by the relationZD ⃗X(τ)e−1¯h S =ZD ⃗Xs(τ)e−1¯h ˜S .

(3.19)The coefficients of |ω||⃗˜X(ω)|2, ωǫσν ˜X∗σ(ω) ˜Xν(ω), and cos Xs(τ)+cos Ys(τ) in ˜S determinethe flows of α, β and V0, respectively. To calculate ˜S, we treat the potential term pertur-batively, just as we did to obtain equation (3.11).

It is not too hard to show that, to firstorder in V0,˜S = 12Z µ−µdω2π˜X†ρ(ω)Sρλ(ω) ˜Xλ(ω) + ⟨SV ⟩f . (3.20)where SV = ΛV0(Λ)Rdτ [cos X(τ) + cos Y (τ)], and the gaussian average over the fastmodes is calculated with the following two-point function:Gρλ(τ) =DXρf (τ)Xλf (0)E=Z Λ−ΛdωS−1ρλ (ω)eiωτW(ω/µ) .

(3.21)W(x) must be ≈0 for x << 1 and ≈1 for x >> 1. To the order in V0 we are calculatinghere, it is sufficient to take W(x) to be a step function.

(If we wish to continue the calcu-lation to higher orders in V , then W(x) must be smooth enough to avoid the generationof spurious long-range behavior in G(τ).) With our choice for W(x), we can calculate thediagonal part of Gρλ(0) with the resultGρλ(0) =2αα2 + β2 ln(Λ/µ)forρ = λ .

(3.22)⟨SV ⟩f can be written solely as a function of GXX(0) = G(0) as follows:⟨SV ⟩f = ΛV0(Λ)e−12 G(0)Zdτ [cos Xs(τ) + cos Ys(τ)] . (3.23)9

Finally, we rescale τ by Λ/µ to restore ˜S to its original form and find that the coefficientof the potential term becomesV0(µ) =V0(Λ)(Λ/µ)e−12 G(0)=V0(Λ)(µ/Λ)(αα2+β2 −1),(3.24)while the coefficients of the friction term and magnetic field term do not change.Weconclude that, as we take µ/Λ to 0, the potential term flows to zero if α/(α2 + β2) > 1,remains fixed when α/(α2 + β2) = 1 and grows when α/(α2 + β2) < 1. Also, to this orderin V0, we have just shown that α and β do not flow.

Therefore, we have demonstratedthe existence of a critical circle in the α-β plane, to first order in V0. We note that weobtain a critical circle for any initial value of V0, as long as V0 is small enough to justifythe perturbative expansion.

Inside this circle, V0 is irrelevant, so the particle should bedelocalized, and, outside the circle, V0 is relevant.We expect this behavior to continue at higher orders in V0 because the only relevantterms that are generated in ˜S are of the form cos Xs(τ) + cos Ys(τ). In particular, thenon-local kinetic term and the magnetic term are not generated, so the friction per unitcell and the magnetic flux cannot flow.

Thus, we expect the circle α2 + β2 to be criticalto all orders in V0.3.3. Symmetries of the Critical TheoryOn the critical circle, whereαα2+β2 =1, the unregulated theory has several nice prop-erties.

First, the dissipative quantum mechanics system displays SL(2, R) invariance, notjust scale invariance. As mentioned in Section 2, we expected this larger symmetry groupat the phase transition because of the connection with open string theory.

This invariancemeans that, under the transformationt →˜t = at + bct + dad −bc = 1a, b, c, d, ∈R ,(3.25)the form of the partition function remains unchanged while ˙X(t) transforms as a dimensionone operator and eikX(t) transforms as a dimension k2 operator.To show this, we first consider the O(V02n) term of the partition function for a neutralgas. For simplicity, we study the zero-field case.

Note, however, that the SL(2, R) invari-ance remains when the magnetic field is non-zero, because the magnetic-field-dependent10

contributions to Zn and to the correlation functions depend only on the ordering of the ti,which remains unchanged under the SL(2, R) transformation.In the zero-field case (β = 0) one has a critical point at α = 1. At this point, we canrewrite equation (3.13) for Z2n asZ2n =1(2n)!V02 e−12⟨X2(0)⟩2n ZnYi=1dti2πdsi2πQi

(3.26)This is the O(V 2n0 ) term of the partition function with +1 charges at the ti’s and −1 chargesat the sj’s. This expression clearly remains unchanged under the translation ti →ti + aand sj →sj + a.

It also remains unchanged when all the variables are rescaled by thesame factor. The third transformation needed to generate SL(2, R) can be taken to bet →−1/t, and it can easily be seen that Z2n is invariant under this inversion as well.

Thus,the partition function is invariant under SL(2, R) transformations.Operator N-point functions are only slightly more complicated.Using the prop-erties of gaussian propagators, one can show that the connected correlation functions⟨˙X(r1) . .

. ˙X(rm)⟩are given by the connected part of the partition function with the in-sertion of(−1)m/2mYj=1(XiD˙X(rj)X(ti)E0 −D˙X(rj)X(si)E0)=(−1)m/2mYj=1(Xi2rj −ti−2rj −si)=(−4)m/2mYj=1(Xiti −si(rj −ti)(rj −si)).

(3.27)Because Z2n is invariant under SL(2, R) transformations, we only need to see how theexpression in curly brackets transforms under rj →˜rj = (arj +b)/(crj +d) with ad−bc = 1(along with a simultaneous transformation of ti and si). We findti −si(rj −ti)(rj −si) →˜ti −˜si(˜rj −˜ti)(˜rj −˜si) = (crj + d)2ti −si(rj −ti)(rj −si) .

(3.28)Taking the derivative of ˜r with respect to r, we haved˜rdr =1(cr + d)2 ,(3.29)from which it follows thatD˙X(r1) . .

. ˙X(rm)E=mYi=1d˜ridri D˙X(˜r1) .

. .

˙X(˜rm)E.(3.30)Thus, insertions of ˙X(r) transform as dimension-1 operators. For correlation functions likeeik1X(t1) .

. .

eik2X(tm)with P kj = 0, the calculation is similar.11

3.4. The Regulated TheorySo far, we have only considered the unregulated theory.

However, it has both infraredand ultraviolet divergences which we must regulate. The ultraviolet divergence was origi-nally regulated by the M ˙X2 term, which acts as a high frequency cutoffby multiplying theFourier-space propagator, ˜G(ω), by 1/(M|ω| + η).

We find it more convenient to use ane−δ|ω| regulator, where δ is a dimensionful ultraviolet cutoff. To take care of the infrareddivergence, we put the particle on a circle of circumference T, which, in Euclidean space,is equivalent to looking at finite temperature.

In this case, the propagator isGµν(t1 −t2) =αα2 + β2Xm̸=01|m|eim(t1−t2)2π/T e−ǫ|m|= −αα2 + β2 ln1 + e−2ǫ −2e−ǫ cos 2πT (t1 −t2)(3.31)when µ = ν andGµν(t1 −t2) = −ǫµνβα2 + β2Xm̸=01meim(t1−t2)2π/Te−ǫ|m|=2iǫµνβα2 + β2 arctansin[2π(t1 −t2)/T]cos[2π(t1 −t2)/T] −eǫ(3.32)for µ ̸= ν.In these expressions, ǫ is a dimensionless cutoffwhich is the ratio of theultraviolet cutoffδ to the infrared scale T. The expressions for Gµν take on much simplerforms if we change variables to zj = e2πitj/T . With this definition, the exponentiatedpropagator ise−qµ1 qν2 Gµν(t1−t2) =−eǫz1z2(z1 −eǫz2)(z1 −e−ǫz2)ζα z1 −z2eǫz2 −z1eǫ · z1z2ζβ,(3.33)where the exponents areζα = −⃗q1 · ⃗q2αα2 + β2 ,ζβ = −qµ1 qν2ǫµνβα2 + β2 ,(3.34)and we have used the identityarctan x = 12i ln1 + ix1 −ix,(3.35)to rewrite the off-diagonal part of G. Using the same variables we also haveD˙X(t1)X(t2)E0 =αα2 + β2−2πiTz21 −z22(z1 −e−ǫz2)(z1 −eǫz2) ,(3.36)12

and we can obtain similar expressions for ⟨˙Y (t1)Y (t2)⟩0, ⟨˙X(t1)Y (t2)⟩0, etc.Now we exploit simplifications which occur on the critical circle, i.e. whenαα2+β2 = 1.First, we note that as ǫ →0 the exponentiated propagator becomese⟨X(t1)X(t2)⟩= −z1z2(z1 −z2)2 .

(3.37)This is very similar to the expression for e⟨X(t1)X(t2)⟩when t lies on a line. Therefore, wecan give a similar argument to the one illustrating SL(2, R) invariance on a line to showthat on the circle, the theory has SU(1, 1) invariance.

This means that underz = e2πit/T →˜z = az + b¯bz + ¯awith|a|2 −|b|2 = 1 ,(3.38)the partition function remains invariant, ˙X(t) transforms as a dimension one operator andso on. As a result, even though at finite temperature the system does not exhibit scaleinvariance, it still possesses a larger symmetry group, SU(1, 1).In order to consider what happens when ǫ ̸= 0, we begin by specializing to the casewhen ⃗qi · ⃗qjαα2+β2 and qµi qνj ǫµνβα2+β2 are integers for all ⃗qi.This condition occurs onthe critical circleαα2+β2 = 1 at the “special points” where β/α ∈Z.

At these points,the partition function and correlation functions have a very simple form even before wetake the regulator to zero. This is because eqµi qνj ⟨Xµ(ti)Xν(tj)⟩and ⟨˙Xµ(ti)Xν(tj)⟩are justrational functions in the zi’s with coefficients of the form enǫ.

For example, at the zero-fieldcritical point, where α = 1 and β = 0, the regulated version of the integral in (3.26) whichgives the O(V 2n0 ) term of the partition function isQ2n =InYj=1dzj2πidwj2πiI2n ,(3.39)where the definition of I2n isI2n = (−1)nQi

Furthermore, thedenominator of the integrand is factored into terms that are linear in all the integrationvariables, zi and wj. Consequently, after we perform the contour integral over one variable,the resulting integrand is once again a rational function in the remaining variables with a13

denominator that is completely factored into linear terms. This property of the integrandhas several important consequences.The first conclusion is that it is straight-forward, although tedious, to analyticallyevaluate any term in the perturbation expansions for the free energy and correlation func-tions.

More generally, there is a well-defined set of rules for integrating any term in theperturbation series. They tell us how to go from a graph (or integrand) with n vertices toseveral graphs with n −1 vertices by evaluating the residues of a rational function.

Withthese rules, we can program a computer to calculate correlation functions analytically. Wehave done a few examples with the help of Mathematica, and some of the results are givenin Section 4.We can also use these rules to prove that the free energy and correlation functionshave certain properties.

The most important such property is that there are no logarithmicdivergences, which can be seen as follows. After we have performed all the integrals, wemust obtain a rational function in eǫ for the free energy (or a rational function of thevariables eǫ and e2πirj for correlation functions of arbitrary numbers of ˙X(rj) and eiX(rj)fields).

This implies that, in the limit as ǫ →0, we obtain only rational functions in ǫ.Thus it is obvious that, to all orders in the periodic potential (at the special points), nologarithmic divergences in ǫ are possible. Pole divergences can of course still occur.This result is actually quite general.

The structure of the partition function integrandwill be the same as above for any doubly periodic potential of the formV = VX cos 2πXa+ VY cos 2πYa+ higher harmonics(3.41)at the special pointsαα2+β2 = 1 andβα2+β2 ∈Z. It appears that imposing the discretesymmetry X →X + na2π , Y →Y + ma2π guarantees that there are no logarithmic divergencesat these points in the phase diagram.

For the general problem of open string tachyonscattering, others [13] have found logarithmic divergences, but here we are looking only atexceptional values of momenta for which their integrals are ill-defined. One of the thingswe have accomplished here is to provide a regularization scheme for which calculations atthese usually singular values of momenta are possible.A careful analysis of the graphs shows that for the case at hand, the graphs for thefree energy with total charge 0 (i.e.P qXj +P qYj = 0) diverge at most as 1/ǫ while allother connected graphs are finite.

Naive power-counting would predict that the charge-onegraphs should diverge logarithmically (which is equivalent to saying that the β-function14

for eiX(t) does not vanish). Quite the contrary, we have proved here that the β-functionfor eiX(t) vanishes at these special points.

In other work, we have also shown that thesetheories exactly satisfy the infinite set of Ward identities predicted by the connection withstring theory outlined in Section 2. An example is included in Appendix B, but the detailsof this statement must await another paper.This general analysis cannot say very much about the points in the phase diagramwhereβα2+β2 /∈Z, but in Section 5 we will offer some evidence in support of the claim,suggested by the results of our renormalization group calculation in Section 3.2, that thethere is a circle of critical theories in the phase diagram to all orders in the periodicpotential.4.

The Free Energy and Correlation Functions for Zero Magnetic FieldWe will now use the methods of the previous section to explicitly calculate the freeenergy and correlation functions at the critical point with β = 0 and α = 1. The mobility,or two-point function, at this critical point has been calculated by Fisher and Zwerger [12]for a weak potential and by Guinea et al.

[14] for the tight-binding model. We provide anew calculation here for the weak potential case which is in agreement with the previousones, and extend it to all N-point functions.

Our method has the advantage that it isfree of approximation and works at some other points on the critical circle with non-zeromagnetic field. In addition, it can be used to calculate correlation functions in the chargedsector, which are of interest in determining the renormalization-group flow and can be usedto calculate the boundary state in open string theory.Even though we can perform any integral in the perturbation series for the free energyand correlation functions, the one draw-back with our choice of regulator is that calcu-lations with it become quite time-consuming at higher orders in V0.

To the order in V0that we have calculated, the main features of the β = 0 neutral sector which emerge arethat: the free energy is proportional to 1/ǫ+O(ǫ) (the significance of this will be explainedSection 4.1); ⟨˙X(t) ˙X(0)⟩is proportional to its value at V0 = 0; all higher m-point func-tions of˙X are zero except for possible contact interactions; and in the limit ǫ →0 allthe correlation functions are SU(1, 1) invariant. For both the neutral and charged graphs,we find that the only needed renormalizations are the subtraction of an infinite constantfrom the free energy and a rescaling of the periodic potential strength.

For our choice ofpotential, this rescaling is equivalent to holding V0Tǫ finite. We will now describe some ofthe calculations which lead to these conclusions.15

4.1. The Free EnergyIf we follow the steps described in Section 3 to introduce convenient ultraviolet andinfrared cutoffs, the partition function (3.26) can be rewritten asZ =Xn1(2n)!

"V0T22(eǫ + e−ǫ −2)#n 2nnQ2n ,(4.1)where Q2n is the integral defined in (3.39).The factor eǫ + e−ǫ −2 comes from self-contractions of X(t) with itself within a single potential insertion e±iX, and the factor ofT multiplying V0 comes from the change of variables from ti to zi. Performing the integralsand expanding in powers of ǫ, we have found thatQ2 =1eǫ −e−ǫ = 12ǫ −112ǫ +7720ǫ3 + O(ǫ5) ,(4.2)Q4 =12ǫ2 −732ǫ −16 + 25192ǫ + O(ǫ2) ,(4.3)andQ6 =34ǫ3 −6364ǫ2 + 716165536ǫ + 34 + O(ǫ) .

(4.4)Putting (4.1) through (4.4) together, we can obtain the first few terms in an expansion ofthe free energy, F = −1T ln Z, in powers of the potential strength:F = −1TV0Tǫ22 12ǫ + O(ǫ)−1TV0Tǫ24 −7128ǫ + O(ǫ)−1TV0Tǫ26 10579786432ǫ + O(ǫ)+ · · · . (4.5)Since ǫ is the dimensionless ratio of the ultraviolet cutoffδ and the infrared cutoffT, itis useful to define a dimensionless potential strength, Vr = V0δ = V0Tǫ, which makes noreference to the infrared scale T. One of our main points will be that the large-T limitdefines a critical theory, independent of the short-distance scale δ, once we have rescaled thepotential strength in this way.

The free energy is then a power series in ǫ with coefficientswhich are themselves power series in Vr:F = −1T∞Xn=−1fn(Vr)ǫn . (4.6)In the case at hand, the f0 term happens to vanish, but we do not expect this to be ageneral feature of the free energy.16

It is instructive to re-express the free energy expansion in terms of the dimensionalscales δ and T:F = −f−1(Vr)1δ −f0(Vr) 1T −f1(Vr) δT 2 + · · ·. (4.7)The only universal term is the f0 term: all the others depend on δ and hence on thedetails of the definition of the theory at short distances.

Furthermore, the f−1 term can becompletely removed by an appropriate Vr-dependent constant shift of the original periodicpotential (such a shift obviously has no observable effect on the particle dynamics). Themeaningful critical physics of the free energy is therefore entirely contained in f0(Vr): Inthe string theory context, it provides the overall normalization of the boundary state (see[9] for examples).

There is also a more conventional thermodynamic interpretation: Insetting up our functional integral in Section 2, instead of integrating out the oscillators, wecould have shown that (2.1) makes the oscillators equivalent to a free massless scalar fieldliving on a one-dimensional world with a boundary (basically a spatial slice of the stringworldsheet) and with some non-trivial, but perfectly local, boundary action for the masslessscalar field induced by the action of the original quantum particle. The path integral overpaths periodic in Euclidean time with period T then generates the partition function of thissystem at temperature τ = 1/T.

Equation (4.7) can then be interpreted as the expansionabout zero temperature of the thermodynamic free energy and f0 can be identified as thezero-temperature limit of the entropy associated with the boundary. (Affleck and Ludwig[15] have recently calculated the boundary entropy of Ising and Kondo systems, and theirpaper gives a clear explanation of how to separate boundary from bulk contributions tothe entropy.) In some sense, this entropy counts the number of dynamically active degreesof freedom living on the boundary.

It should (and does) vanish for dynamically trivialboundary conditions like Dirichlet or Neumann. What it should do for the case at hand,where there is a complicated boundary action, is not obvious.

The calculations we havesummarized in (4.5) show, perhaps surprisingly, that f0(Vr) is identically zero. We willlater see that, if the magnetic field is turned on as well as the potential, f0 becomes anon-zero function of the magnetic field and of Vr.

In view of this discussion, in the rest ofthis paper we shall identify the free energy with the f0 term in the appropriate analog ofthe expansion (4.7). It would be more accurate to speak of the zero-temperature entropy,but this abuse of language should cause no confusion.17

4.2. The N-Point FunctionsNext, we turn our attention to the m-point functions.

When m is odd, the m-pointfunctions are zero by symmetry.For the connected 2m-point functions, we insert theregulated version of (3.27) into the partition function, and absorb the self contractions ofthe potential insertions into Vr. The result is that we must compute the connected part ofD˙X(r1) .

. .

˙X(r2m)E=D˙X(r1) . .

. ˙X(r2m)E0 +∞Xn=1(Vr/2)2n(2n)!2nn 2πiT2mA2n .

(4.8)Here, ⟨˙X(r1) . .

. ˙X(r2m)⟩0 is the 2m-point function in the absence of the periodic potentialand A2n is defined to beA2n =InYj=1dzj2πidwj2πi I2nRn,m ,(4.9)where I2n is the function defined in (3.40) andRn,m = (−1)m2mYj=1nXi=1 ξ2j −z2i(ξj −e−ǫzi)(ξj −eǫzi) −ξ2j −w2i(ξj −e−ǫwi)(ξj −eǫwi)!,(4.10)with ξj = e2πirj/T .For the two-point function, we have done explicit calculations out to fourth order inthe potential, findingA2 = −12 sin2 2πT t1−t22 + O(ǫ2) ;(4.11)andA4 =−2ǫ + 312 sin2 2πT t1−t22 .

(4.12)These two expressions are both proportional to the two-point function in the absence ofthe periodic potential, which is given byD˙X(t1) ˙X(t2)E0 = −2πT212 sin2 2πT t1−t22 . (4.13)To this order in V0, A2n contains one non-zero disconnected graph, coming from Z2 · A2.When we substitute the expression for A0, A2 and A4 back into equation (4.8) for the2m-point function, and then subtract offthe disconnected graph, we find that as ǫ →0,D˙X(t1) ˙X(t2)Econn = µ(Vr)D˙X(t1) ˙X(t2)E0 ,(4.14)18

where µ is given byµ(Vr) = 1 −Vr22+ 34Vr24+ · · ·. (4.15)Therefore, the renormalized two-point function is equal to the “free” two-point function,rescaled by µ(Vr).For small enough V0, we see that µ < 1 but, lacking an all-orderscalculation, we cannot show that µ is always less than 1.

(In ref. [4], we sketch analternate calculation in which we fermionize the theory.

In that case, we can calculatethe mobility to all orders in Vr and we find that 0 ≤µ ≤1.) We note here that, afterrescaling V0, the answer is finite (except when t1 = t2).

This means that we do not needto renormalize the kinetic part (2.5) of the action, despite the fact that the propagatorcorrection diagrams appeared to have a divergence ∝1/ǫ. These results are consistentwith what was found in the tight-binding approximation some time ago [14].We have also used Mathematica to evaluate the integrals for the four-point function toorder V 40 and have found that the connected four-point function, ⟨˙X(r1) .

. .

˙X(r4)⟩, is zeroas ǫ →0, except when ri = rj. In addition, we have calculated the connected 2m-pointfunctions for any m > 1 at order V 20 and found them to vanish as ǫ →0, as long as noneof the points are coincident.

Both calculations are quite involved and the details of thelatter are given in Appendix A.In summary, the critical theory looks basically like the free theory, except that thetwo-point function is rescaled and the 2m-point functions have contact terms. The criticaltheory is obtained as the large-T (zero-temperature!) limit of the theory with a fixed short-distance scale.

The only needed renormalizations are a rescaling of the potential strength(the critical theory depends only on Vr = V0δ) and the subtraction of a temperature-independent constant from F. The two-point function differs from the free two-point func-tion by a finite mobility factor µ(Vr) which decreases from unity as Vr increases from zero.We expect that the mobility should be less than one for any Vr because the periodic poten-tial should inhibit the particle’s motion. Lastly, the critical theory is SU(1, 1)-invariant:The interacting two-point function is proportional to the free two-point function, which isSU(1, 1)-invariant, and the higher-point functions, barring contact terms, are zero.

(Ac-tually, using the fermionic regulator of [4], we can show that even the contact terms areSU(1, 1)-invariant. )19

5. Magnetic Field EffectsWe now turn to our main problem, that of calculating the critical properties of aparticle moving in two dimensions and simultaneously subject to a magnetic field and aperiodic potential.

As before, we will calculate both the free energy and a variety of N-point functions in a perturbation expansion in the potential strength (but the dependenceon the magnetic field will be exactly accounted for). We will also restrict ourselves to theneutral sector of the Coulomb gas arising from the expansion in powers of the periodicpotential.

In what follows we assume that we are on the critical circleαα2+β2 = 1 andidentify where we are on that circle by the value of γ = β/α. So far we have provedto all orders in V0 that the points where γ ∈Z are exactly critical, and have shown inSection 3.2 that to the first few orders in V0 the theory is also critical for γ /∈Z.

Wenow support this with some explicit calculations, valid for any γ, which again show nologarithmic divergences.5.1. Magnetic Field Contribution to Free EnergyLet us first deal with the free energy, expanding it in powers of the potential strength:F(B, Vr) = −P∞0 Fn(B)V nr .

The term of zeroth-order in Vr has been computed elsewhere[16,17] and has the interesting formF0(B) = 12T ln(1 + (2πα′B)2) . (5.1)As explained in [17] and elsewhere, if this expression is treated as an action functional, itgenerates stringy corrections to Maxwell’s equations.

It is at least an existence proof thatthe boundary entropy discussed in the previous section can have non-trivial dependenceon the parameters of the critical theory. We will not discuss it further, as we are reallyinterested in the joint effects of potential and magnetic field which are responsible for theunusual spectrum of the Hofstadter problem with no dissipation.

Such effects do not occuruntil fourth-order in the expansion in powers of the periodic potential. This is the firstorder at which we can have the insertion of both X and Y “charges”, so that the systemcan feel the effect of the phases due to the magnetic field.

(By the zero-charge condition,each eiX must be accompanied by an e−iX and each eiY by an e−iY .) In what follows, wewill evaluate the free energy to order V 40 (it is possible, but tedious to go to higher orders).For calculations involving non-zero magnetic field, the contour integral technique in-troduced in Section 3 encounters difficulties at fourth order in the potential strength.

This20

is precisely because the interesting interplay between B and V begins at this order, and theorigin of the problem can be seen explicitly in the part of (3.33) which is the exponentialof the off-diagonal propagator. Diagrams with both X and Y internal charges will involveintegrals over expressions containing factors of this kind, and whenever γ is not an integer,branch cuts will be present which make evaluation difficult.

To avoid this complication,we choose to use a new method of calculation starting directly with equation (3.14) for thepartition function. In the unregulated diagrams, the exponentiated off-diagonal propaga-tors which connect X and Y charges simply contribute a phase factor when an X chargemoves past a Y charge.

Except for the above-mentioned V0-independent piece, the mag-netic field only contributes to the free energy through these phases. Thus we will take intoaccount the interaction between the the X and Y charges simply by keeping track of thephase factors.

We will regulate the diagonal propagator in the usual way. Unfortunately,the necessity to maintain the ordering of the charges makes it impossible to use contourintegration and so we resort to series expansion of the diagonal propagator instead.

Theadvantage of this technique of course is that it works for non-integer values of γ.One might worry that we are not regulating the sign-function which generates thephases, but this is only of concern in the calculation of correlation functions. If we simplywish to calculate the free energy, the off-diagonal propagator contributes no divergences,and we expect that the phase prescription will work correctly.

In calculating correlationfunctions, however, we take derivatives of the off-diagonal propagator and regulation be-comes necessary because the derivative of a sign-function is a δ-function.As we indicated above, the first B-dependence in the free energy comes at order V 40 ,and we now proceed to calculate this term. The contribution to the partition function ofthe diagram with nX of the X charges and nY of the Y charges is (with n = nX + nY )Z(nX, nY ) =V02 e−12 G(0)nX⃗qi= (±1,0)(0,±1)Zdτ1 .

. .

dτn expXi

the integration range is 0 ≤τi ≤τi+1 ≤T and G(t) is the diagonal element of thepropagator (3.8). The fourth-order piece of the free energy is given byF4 = −4TV02 e−12 G(0)4 e2πiγ + e−2πiγ −2Zdt1dt2ds1ds2 expG(t1 −t2) + G(s1 −s2)(5.4)where: 0 ≤t1 ≤s1 ≤t2 ≤s2 ≤T; the overall factor of 4 takes account of the fact that thefirst charge can be a plus or a minus and an X or a Y ; the −2 in the γ-dependent factorsubtracts the contribution due to the disconnected graphs, which are also present in theabsence of a magnetic field.To get finite results we must, of course, regulate the propagator.

Because we are onthe critical circle, we can use the strategy described in Section 3 following (3.31). Thisamounts to the replacementG(z) = −ln1 −ze−ǫ 1 −¯ze−ǫ(5.5)with z = exp (2πit/T).

For our later calculation it will be useful to have the followingpower series expansion of the exponentiated propagatoreG(z) =11 −e−2ǫ∞Xm=−∞zme−|m|ǫ . (5.6)For coincident points this reduces toe−G(0) =1 −e−ǫ2 .

(5.7)At this point we note the translational invariance and periodicity of G and use the formulaZ T0dτ1 . .

. dτnn!f(τi −τj) = TnZdτ2 .

. .dτnf(τi −τj)τ1=0(5.8)where on the right hand side 0 ≤τi ≤τi+1 ≤T and f(τ) is periodic in all its argumentswith period T. This allows us to writeF4 = 4 sin2 πγV02 e−12 G(0)4 Zds1dt2ds2expG(t2) + G(s1 −s2)(5.9)22

where again 0 ≤s1 ≤t2 ≤s2 ≤T. Transforming to angular variables on a circle and using(5.6) we obtainF4 = 4 sin2 πγV02 e−12 G(0)4 T2π3 11 −e−2ǫ2∞Xm,n=−∞e−|m|ǫe−|n|ǫImn(5.10)where the integral is given byImn =Z 2π0dφ2Z φ20dθ2Z θ20dφ1eimθ2ein(φ2−φ1) .

(5.11)Doing the integral we findImn = 4π33 δm,0δn,0 −4πm2 δn,0 (1 −δm,0) −4πn2 δm,0 (1 −δn,0)+2πn2 (1 −δn,0) (δm,n + δm,−n)(5.12)and therefore thatXm,ne−|m|ǫe−|n|ǫImn = 4π33−8π∞Xm=11m22e−mǫ −e−2mǫ. (5.13)The sum can be evaluated for small ǫ and we find that∞Xm=11m22e−mǫ −e−2mǫ= π26 −2ǫ ln 2 + ǫ22 −ǫ312 + O(ǫ4) .

(5.14)Finally, using the renormalization of the potential strength introduced in the previoussection, the free energy becomesF4 =V 4r32π2T sin2 πγ4 ln 2ǫ−1 + O(ǫ). (5.15)The ǫ−1 term must of course be subtracted away, but a finite magnetic-field-dependentpiece is left over.This finite part vanishes for any integer γ = k.This is physicallyreasonable since, at these points, the magnetic phase associated with transporting thecenter of the electron’s orbit around a unit cell of the reciprocal lattice (as defined inequation (3.15)) is a multiple of 2π.

Therefore we expect this situation to look equivalentto the zero field case. Consideration of the phase prescription for higher orders in thepotential shows that all such terms will also be zero for these special points on the criticalcircle.

Of particular importance is the observation that there is no logarithmic divergenceeven when γ is not an integer. If this remains true for higher orders, as we believe it will,then it confirms the result of Section 3.2 that we have a critical circle, α = α2 + β2.23

5.2. N-Point Functions and Magnetic FieldWe now turn to the calculation of the m-point functions on the critical circle.

Again,the (2m + 1)-point functions are all trivially zero by symmetry arguments. We presentexplicit calculations of the 2m-point functions to O(V 20 ) using the contour integral tech-niques outlined in Section 3.

Our results are valid for any value of γ, not just integer ones.As we saw above, the extension of these results to higher orders in the strength of theperiodic potential is more troublesome because, unless γ is an integer, there are branchcuts in the integrands whenever there is an internal line connecting internal X and Ycharges. However, our earlier calculations show that when γ is an integer, nothing remark-able happens at higher orders, and certainly that there will be no logarithmic divergences.For non-integer γ, the presence of the branch cuts makes it harder to be certain, but theabsence of logarithmic divergences in the free energy is strong evidence that even here theO(V 20 ) behavior will carry on essentially unchanged at higher orders.The O(V 20 ) contribution to the 2m-point functions has the formD˙Xµ1(r1) .

. .

˙Xµ2m(r2m)E2 =V02 e−12 G(0)2 Zdt1dt2 expGXX(t1 −t2) RX(2m) + RY (2m),(5.16)whereRν(2m) = (−1)m2mYi=1 ddriGµiν(ri −t1) −ddriGµiν(ri −t2). (5.17)RX(2m) comes from two cos X potential insertions, and RY (2m) comes from two cos Ypotential insertions.

Now let G be equal to GXX and N equal to GXY , where the propa-gators have been regulated as in Section 3. Then, with the definition zj = exp(2πitj/T),we have∂1G(t1 −t2) = 2πiTz2z2 −e−ǫz1−z1z1 −e−ǫz2((5.18)(5.18)equation5.18equation5.185.18)and∂1N(t1 −t2) = γ 2πiTz1z1 −eǫz2+z2z2 −eǫz1.

((5.18)(5.18)equation5.19equation5.195.19)We will first restrict our attention to the calculation of the ⟨˙X ˙X⟩and ⟨˙X ˙Y ⟩two-pointfunctions. Because our system is invariant under rotations in the X-Y plane, ⟨˙Y ˙Y ⟩has(5.18)(5.18)equation5.20page2424

the same form as ⟨˙X ˙X⟩. The integrals for the second order contributions to the two-pointfunctions can be written asD˙X(t1) ˙X(t2)E2 = −V02 e−12 G(0)2 Z T0dudv eG(u−v)Iα(u, v)((5.18)(5.18)equation5.20equation5.205.20)andD˙X(t1) ˙Y (t2)E2 = −V02 e−12 G(0)2 Z T0dudv eG(u−v)Iβ(u, v)((5.18)(5.18)equation5.21equation5.215.21)where Iα is defined byIα(u, v) = 2∂1G(t1 −u)∂2G(t2 −u) −G(t2 −v)+ 2∂1N(t1 −u)∂2N(t2 −u) −N(t2 −v),((5.18)(5.18)equation5.22equation5.225.22)and Iβ is given byIβ(u, v) = 2∂1G(t1 −u)∂2N(t2 −v) −N(t2 −u)+ 2∂1N(t1 −u)∂2G(t2 −u) −G(t2 −v).

((5.18)(5.18)equation5.23equation5.235.23)The calculation by contour integration is straight-forward and gives for the diagonal con-tributionD˙X(t1) ˙X(t2)E2 =Vr22 2πT2(1 −γ2)h2 sin2 πT (t1 −t2)i−1= −(1 −γ2)Vr22 D˙X(t1) ˙X(t2)E0((5.18)(5.18)equation5.24equation5.245.24)where Vr is the renormalized potential, given by Vr = V0Tǫ. This two-point function isthe same as the free propagator with a rescaled coefficient depending on the renormalizedpotential strength and the magnetic field.For the off-diagonal part we get zero for separated points and the contact term lookslike the derivative of a delta function, or two derivatives of a step function, as we wouldexpect if this were also just a rescaled version of the free off-diagonal propagator.

Explicitly,we findD˙X(t1) ˙Y (t2)E2 = −iπγV 2r δ′(t1 −t2) . ((5.18)(5.18)equation5.25equation5.255.25)(5.18)(5.18)equation5.26page2525

Integration of the regulated version of this result, or a direct calculation using the seriesexpansion (5.6) for G (and a similar one for N) gives the result that⟨X(t1)Y (t2)⟩2 = −2πiγVr22 2T (t1 −t2) −sign(t1 −t2)= −2Vr22⟨X(t1)Y (t2)⟩0 . ((5.18)(5.18)equation5.26equation5.265.26)The constants of integration which arise upon integrating (5.25) twice to obtain (5.26) arefixed by the requirement that all functions be periodic with period T. Thus the secondorder contribution is just −2(Vr/2)2 times the free propagator.

The conclusion here isthat when we turn on the potential in the presence of the magnetic field, the zeroth-ordertwo-point function ⟨˙Xµ ˙Xν⟩is just multiplied by a constant matrix whose elements arefunctions of γ and Vr. For the special points in the phase diagram where γ is integer weknow for sure that this remains true at higher orders in the potential, but we have notproved this when γ is not an integer.In reference [4], we have shown that the correlation functions on the critical circle atinteger γ are all related by a duality symmetry.

Given the form of the two-point functionfor γ = 1, this symmetry completely determines the form of the two-point functions at allthe other special critical points. It is interesting to compare the duality prediction with theexplicit functions we have just calculated.

To do this, it is convenient to specify locationin the α-β plane by the complex number z = α + iβ and to rewrite the free two-pointfunction (3.8) as follows:Gµν(t; z) = −Re(1/z) ln(t2) δµν + iπ Im(1/z) sign(t) ǫµν. ((5.18)(5.18)equation5.27equation5.275.27)(For simplicity we have removed the infrared cutoffand rewritten this as a two-pointfunction on the open line.) The content of the duality relation derived in [4] is that we canexpress the interacting two-point function at the special points on the critical circle solelyin terms of the function G at various values of z, and of the zero-magnetic-field mobilityfunction, µ(Vr), displayed in (4.15).

Define zγ = 1/(1 −iγ). Then when α/(α2 + β2) = 1and β/α = γ, we have z equal to zγ.

According to equation 6.2 in [4], when γ is an integer,the two-point function with potential strength V0 is given byD⃗X(t) ⃗X(0)E(γ, V0) = G(t; zγ) + (µ(Vr) −1) G(t; zγ2) . ((5.18)(5.18)equation5.28equation5.285.28)(5.18)(5.18)equation5.29page2626

The first term on the right-hand side is just the expression for the two-point function inthe absence of the potential. The second term contains all the effects due to the potential.We note that 1/zγ2 = (1 −γ2) −2iγ, and that [µ(Vr) −1] = −(Vr/2)2 + O(V 4r ).

Puttingthis together with (5.27), we see that (5.28) predicts the order V 20 part of the two-pointfunction to be⟨Xµ(t)Xν(0)⟩(γ, V0) =Vr22 1 −γ2ln(t2) δµν+ iπVr222 γ sign(t) ǫµν. ((5.18)(5.18)equation5.29equation5.295.29)This can be expressed in terms of the free two-point function (5.27) at γ as follows:D˙X(t) ˙X(0)E2 (γ, V0) = −(1 −γ2)Vr22 D˙X(t) ˙X(0)E0 ,((5.18)(5.18)equation5.30equation5.305.30)andD˙X(t) ˙Y (0)E2 (γ, V0) = −2Vr22 D˙X(t) ˙Y (0)E0 .

((5.18)(5.18)equation5.31equation5.315.31)On comparing this expression with equations (5.24) and (5.26), we see that it agrees withour direct computation. We conclude that not only do the two-point functions exhibitSL(2, R) covariance as a function of the time variable, but they also satisfy a dualitysymmetry that relates two-point functions at different values of friction and flux.For the 2m-point functions with m ≥2, we can again perform the contour integrals andsum over all the diagrams, exactly as we do in Appendix A for the 2m-point functions withzero magnetic field.

This time, we find that not all the correlation functions vanish whenthe points are non-coincident. To this order in the potential, any correlation function withexactly two ˙X’s or two ˙Y ’s is finite as the cut-offgoes to zero and all the other correlationfunctions are zero except for contact terms.

When there are exactly two ˙X’s, the result isD˙X(t1) ˙X(t2) ˙Y (s1) . .

. ˙Y (s2m)E2 = Cm2mYj=1z1 + wjz1 −wj−z2 + wjz2 −wjz1z2(z1 −z2)2 ,((5.18)(5.18)equation5.32equation5.325.32)for m ≥2, and similarly when there are exactly two ˙Y ’s.

In this equation, we have definedCm =2πT(2m+2) Vr22γ2 ,((5.18)(5.18)equation5.33equation5.335.33)(5.18)(5.18)equation5.34page2727

zj = e2πitj/T and wj = e2πisj/T . The four-point function can be written more simply asD˙X(t1) ˙X(t2) ˙Y (s1) ˙Y (s2)E2 = C116z1z2w1w2(z1 −w1)(w1 −z2)(z2 −w2)(w2 −z2) .

((5.18)(5.18)equation5.34equation5.345.34)We note that these correlation functions have the same form as the integrand of the O(V 20 )contribution of the 2m-point function with no magnetic field. Using arguments similar tothose in Section 3, we can show that such functions are SU(1, 1) covariant (or SL(2, R)covariant in the limit as T →∞).Thus, we have found non-trivial critical theorieswhich exhibit not only scale invariance, but also the higher symmetry group, SL(2, R),as predicted by their connection with string theory.

At higher orders in V0, when γ isan integer, we expect that additional correlation functions will have a finite, SL(2, R)-covariant limit as the cut-offis taken to zero.(5.18)(5.18)equation6.1section66. Charged Sector DiagramsSo far in this paper we have restricted our attention to the neutral sector of theCoulomb gas arising from the perturbative expansion in the periodic potential.

In theDQM path integral, this restriction is enforced by the integral over the zero-mode. Instring theory, the zero-mode integral usually serves to enforce momentum conservation inS-matrix elements.

However, in calculating the open string boundary state, the zero-modeintegration is left undone, and the charged sector contributes. Charged diagrams are also ofinterest in their own right from a Coulomb gas point of view and are useful in calculatingthe renormalization-group flow (e.g.

as in Section 3.2 or in [13]). We shall see that inthe critical theory, the connected diagrams of the charged sectors are all completely finitefunctions of the rescaled potential Vr.

For simplicity, we restrict ourselves in this sectionto zero magnetic field.(5.18)(5.18)equation6.1subsection6.16.1. Charged Sector Free EnergyAs shown previously for the critical theory at zero magnetic field, all diagrams canbe calculated by contour integration tricks.

The nth order contribution to the partitionfunction is given byZn = 1n!V02 e−12 G(0)nX{qj=±1}eiQnX0Z T/2−T/2nYi=1dti exp−Xj

where Qn = P qj, G(t) is defined by eqn. (5.5) and X0 is the zero-mode.

We have donethe calculation to fourth order in V0 and found the following results for Z (we expresseverything in terms of the usual rescaled potential strength Vr = V0Tǫ):Z0 = 1 ,((5.18)(5.18)equation6.2equation6.26.2)Z1 = 2Vr2cos X0h1 −ǫ2 + O(ǫ2)i,((5.18)(5.18)equation6.3equation6.36.3)Z2 = 2Vr22 14ǫ + (1 −2ǫ) cos 2X0 −ǫ48 + O(ǫ2),((5.18)(5.18)equation6.4equation6.46.4)Z3 = 2Vr23 12ǫ cos X0 + cos 3X0 −58 cos X0 + O(ǫ),((5.18)(5.18)equation6.5equation6.56.5)Z4 = 2Vr24 " 14ǫ2+ 12ǫcos 2X0 −7128−196 + cos 4X0 −53 cos 2X0 + O(ǫ). ((5.18)(5.18)equation6.6equation6.66.6)The free energy is related to the connected part of these diagrams in the usual way:F = −1T ln Z = −2T∞Xn=1Vr2nFn .

((5.18)(5.18)equation6.7equation6.76.7)Rearranging the series for Z, we find that the first four Fn areF1 = cos X0 ,((5.18)(5.18)equation6.8equation6.86.8)F2 = 14ǫ + 12 cos 2X0 −12 ,((5.18)(5.18)equation6.9equation6.96.9)F3 = 13 cos 3X0 −38 cos X0 ,((5.18)(5.18)equation6.10equation6.106.10)F4 = −7256ǫ + 14 cos 4X0 −724 cos 2X0 + 18 . ((5.18)(5.18)equation6.11equation6.116.11)The contribution of the charge-Q diagrams is given by the coefficient of eiQX0 in theseexpressions.All the divergences come from the neutral sector and can be removed bythe subtraction of a Vr-dependent 1/ǫ term.

Using general arguments about the form ofthe diagrams, we can show that this behavior continues to all orders. The key thing tonote here is that the logarithmic divergences expected in such diagrams according to naivepower-counting arguments are completely absent.

Using our method of calculation, this isof course guaranteed from the beginning. (5.18)(5.18)equation6.12page2929

(5.18)(5.18)equation6.12subsection6.26.2. Charged Sector Two-Point FunctionCalculation of the N-point functions can be done in a very similar way.

The two-point function is particularly important for the open string boundary state (one extractsthe energy-momentum tensor from it) and also for checking consistency with the Wardidentity, so we show the calculation here as an example. The explicit formula for the n-thorder contribution to the two-point function isD˙X(τ1) ˙X(τ2)En = −1Z1n!V02 e−12 G(0)n Xqj=±1eiQnX0nXi,j=1qiqjIij(n, ⃗q)((5.18)(5.18)equation6.12equation6.126.12)where we define Iij(n, ⃗q) byIij(n, ⃗q) =Zdt1 .

. .

dtn ∂1G(τ1 −ti) ∂2G(τ2 −tj)nYk=2k−1Yl=1expqkqlG(tk −tl). ((5.18)(5.18)equation6.13equation6.136.13)Explicit results for the first three terms in an expansion in powers of the potential areD˙X(τ1) ˙X(τ2)E0 = −2πT2 2 sin2 πT τ12−1,((5.18)(5.18)equation6.14equation6.146.14)D˙X(τ1) ˙X(τ2)E1 = Vr cos X02πT2 1 −∆2ǫ2πT τ12,((5.18)(5.18)equation6.15equation6.156.15)D˙X(τ1) ˙X(τ2)E2 =Vr22 2πT2 8 cos 2X0 cos2 πT τ12−4 cos2 X0+4 sin2 X0 ∆2ǫ2πT τ12+2 sin2 πT τ12−1,((5.18)(5.18)equation6.16equation6.166.16)where ∆is a regulated delta-function defined by∆nǫ(τ) = limǫ→0sinh(nǫ)cosh(nǫ) −cos τ((5.18)(5.18)equation6.17equation6.176.17)and τ12 = τ1−τ2.

Note that while all divergences are eliminated by rescaling the potential,the answer contains contact terms.An important aspect of this result is that the zero-mode-dependent pieces are notSU(1, 1)-invariant. This has to do with the fact, explained in [10], that reparametrizationinvariance of the boundary state path integral is not manifest: it is in some sense “softly(5.18)(5.18)equation6.18page3030

broken” by the non-local kinetic term in the boundary state action and the fixing of thezero mode in the boundary state path integral measure. There is nonetheless a completeset of “broken reparametrization invariance” Ward identities in which all of these effectsare included [10].

As it turns out, the full Ward identity for the zero-mode-dependenttwo-point function involves the zero-mode-dependent part of the the free energy as well.The calculations reported in this section can therefore be used to make a rather nontrivialcheck of the Ward identities (not to speak of our whole renormalization scheme). We havechecked that the above expressions satisfy the identity.

The presence of the contact terms(terms involving ∆) in the two-point function turns out to be a crucial element in theconsistency of the Ward identity (details are relegated to an appendix). We conclude thatour renormalization scheme produces a genuine solution to open string theory.Given these results we can also explicitly construct the tachyon and massless particlecontributions to the boundary state, defined in equation (2.3), up to second order in thetachyon strength, V0.From the massless contribution, space-time equations of motioncan be derived for the graviton and dilaton in the presence of this background field, andthe stress energy tensor in Einstein’s equations due to the tachyon source can be con-structed explicitly.

However, efforts to discover a space-time effective action from whichthe equations of motion could be derived were unsuccessful, probably because the tachyonfluctuates on a short scale, rendering meaningless a low-energy description of its effects.(5.18)(5.18)equation7.1section77. ConclusionsThe state of affairs described in this paper is promising, but far from fully satisfactory.Our goal is to find as complete a characterization of the “Hofstadter” critical theories as onehas for the Ising model or the WZW models.

The purely perturbative approach describedin this paper is obviously not going to take us that far, but it does give a pretty clear ideaof what we would eventually like to prove. Since one of the marginal terms (the dissipationterm) in the action is non-local, standard field theory intuition does not necessarily applyand we have had to invent special methods and re-examine the whole question of criticalbehavior from the ground up.

Our primary claim, supported by a variety of perturbativecalculations, is that there is a two-parameter critical surface: The starting action hasthree marginal parameters (dissipation constant, magnetic field strength and strength ofthe periodic potential) and we have offered evidence that there is a critical theory forany value of the renormalized potential strength so long as the dissipation constant and(5.18)(5.18)equation7.1page3131

magnetic field satisfy one functional condition. (Basically the periodic potential has tobe a dimension-one operator in the one-loop approximation!

This is reminiscent of thesituation in the Liouville theory.) We have also calculated some of the critical N-pointfunctions.

Because we are working in the rather unfamiliar context of critical theorieswith nonlocal interactions, we have found it worthwhile to use these N-point functionsto explicitly check the reparametrization invariance Ward identities (this ensures that thetheory is not just critical, but also a solution of string theory, a much more stringentrequirement). A further, surprising, finding is that many of the higher N-point functions(all of them in the case of zero magnetic field) reduce to contact terms.

This means thatthese critical theories are almost, but not quite, trivial. This is a broad hint that an exactsolution for this system should be possible, but we have yet to see how to exploit it.

Wehope to return to this point in a future publication.AcknowledgementsThis work was supported in part by DOE grant DE-AC02-84-1553. D.F.

was alsosupported by DOE grant DE-AC02-76ER03069 and by NSF grant 87-08447. A.F.

alsoreceived support from an Upton Foundation Fellowship.Appendix (5.18)(5.18)equation7.1appendixAA. N-Point FunctionsIn this appendix, for zero magnetic field, we evaluate the O(V 20 ) contribution to thehigher 2m-point functions of ˙X and demonstrate that, except for contact terms, they arezero as ǫ →0.

As described in Section 4, the integral we want to evaluate is A2 (4.9):A2 =I dz12πidz22πiI2R1,m ,((5.18)(5.18)equationA.1equationA.1A.1)withI2 = −1(z1 −e−ǫz2)(z1 −eǫz2)((5.18)(5.18)equationA.2equationA.2A.2)andR1,m =(−1)m2mYj=1 w2j −z21(wj −e−ǫz1)(wj −eǫz1) −w2j −z22(wj −e−ǫz2)(wj −eǫz2)!=(−1)m2mXM=0XσMYj∈σMw2j −z21(wj −e−ǫz1)(wj −eǫz1)×Yi/∈σMw2i −z22(wi −e−ǫz2)(wi −eǫz2)(−1)M ,((5.18)(5.18)equationA.3equationA.3A.3)(5.18)(5.18)equationA.4page3232

where σM is summed over all subsets of the first 2m integers which contain M elements.We will proceed by performing the integralI dz12πidz22πiMYj=1w2j −z21(wj −e−ǫz1)(wj −eǫz1)2mYi=M+1w2i −z22(wi −e−ǫz2)(wi −eǫz2)I2((5.18)(5.18)equationA.4equationA.4A.4)and then symmetrize later. If we first perform the z1 integral, we must evaluate the residueswhen z1 = e−ǫz2 and, if M > 0, also when z1 = e−ǫwk, for 1 ≤k ≤M.

We will call theresidue from the z1 = e−ǫz2 pole rz, and the residues from the z1 = e−ǫwk poles rk. Theyare given byrz =1z2(eǫ −e−ǫ)MYi=1(z22e−ǫ −w2i eǫ)(z2 −wi)(z2e−ǫ −wieǫ)2mYj=M+1z22 −w2j(z2 −e−ǫwj)(z2 −eǫwj) ,((5.18)(5.18)equationA.5equationA.5A.5)andrk =wk(wk −z2)(eǫz2 −e−ǫwk)MYi=1i̸=kw2ke−ǫ −w2i eǫ(wke−ǫ −wieǫ)(wk −wi)×2mYj=M+1z22 −w2j(z2 −e−ǫwj)(z2 −eǫwj) ,((5.18)(5.18)equationA.6equationA.6A.6)for 1 ≤k ≤M.rk and rz both appear to have poles on the z2 contour when z2 = wk or wi, respectively.However, we started with a completely well-defined, convergent integral for any value ofz2 and wi, so we know that the result, r2 + Pk rk, must be finite.

Consequently, all thepoles on the z2 contour must cancel each other when we add up all the residues. (We havechecked this explicitly for m ≤2.) Therefore, we can integrate each residue separately andjust ignore the poles on the contour.

Then, forHrzdz22πi, we must evaluate residues whenz2 = 0 and z2 = e−ǫwl for M + 1 ≤l ≤2m. The residue when z2 = 0 isrzz =1eǫ −e−ǫ .

((5.18)(5.18)equationA.7equationA.7A.7)The residue when z2 = e−ǫwl isrzl =1eǫ −e−ǫMYi=1w2l e−2ǫ −w2i e2ǫ(wle−ǫ/2 −wieǫ/2)(wle−3ǫ/2 −wie3ǫ/2)×2mYj=M+1j̸=lw2l e−ǫ −w2j eǫ(wl −wj)(wle−ǫ −wjeǫ) . ((5.18)(5.18)equationA.8equationA.8A.8)(5.18)(5.18)equationA.9page3333

The integralHrkdz22πi has poles inside the contour when z2 = e−2ǫwk and z2 = e−ǫwlfor M + 1 ≤l ≤2m. The residue when z2 = e−2ǫwk isrkk =1eǫ −e−ǫMYi=1i̸=kw2ke−ǫ −w2i eǫ(wke−ǫ −wieǫ)(wk −wi)×2mYj=M+1w2ke−2ǫ −w2j e2ǫ(wke−ǫ/2 −wjeǫ/2)(wke−3ǫ/2 −wje3ǫ/2) ;((5.18)(5.18)equationA.9equationA.9A.9)and, lastly, the residue when z2 = e−ǫwl for M + 1 ≤l ≤2m isrkl =wkwl(wkeǫ/2 −wle−ǫ/2)(wleǫ/2 −wke−ǫ/2)MYi=1i̸=kw2ke−ǫ −w2i eǫ(wke−ǫ −wieǫ)(wk −wi)×2mYj=M+1j̸=lw2l e−ǫ −w2jeǫ(wl −wj)(wle−ǫ −wjeǫ) .

((5.18)(5.18)equationA.10equationA.10A.10)The total 2m-point function is then given by a sum over all the residues, symmetrizedin the wk’s:A2 = (−1)m2mXM=0XσM(−1)Mrzz +Xl/∈σMrzl +Xk∈σMrkk +Xk∈σMXl/∈σMrkl. ((5.18)(5.18)equationA.11equationA.11A.11)To evaluate A2 as ǫ →0, we Taylor expand rzz, rzl, rkk and rkl.

After some algebra,we findrzz =1eǫ −e−ǫ = 12ǫ + O(ǫ) ;((5.18)(5.18)equationA.12equationA.12A.12)rzl = 12ǫf(wl) + 2Xp∈σMh(wl, wp) +Xp/∈σMp̸=lh(wl, wp) + O(ǫ) ;((5.18)(5.18)equationA.13equationA.13A.13)rkk = 12ǫf(wk) +Xp∈σMp̸=kh(wk, wp) + 2Xp/∈σMh(wk, wp) + O(ǫ) ;((5.18)(5.18)equationA.14equationA.14A.14)andrkl = −HσM (k, l) ,((5.18)(5.18)equationA.15equationA.15A.15)(5.18)(5.18)equationA.17page3434

where we have definedf(wl) =2mYi=1i̸=lwl + wiwl −wi;((5.18)(5.18)equationA.16equationA.16A.16)h(wk, wp) =wkwp(wk −wp)22mYi=1i̸=k,i̸=pwk + wiwk −wi; ((5.18)(5.18)equationA.17equationA.17A.17)andHσM (k, l) =wkwl(wk −wl)2Yi∈σMi̸=kwk + wiwk −wiYj /∈σMj̸=lwl + wjwl −wj. ((5.18)(5.18)equationA.18equationA.18A.18)We will also define the functionsF =2mXj=1f(wj) ;((5.18)(5.18)equationA.19equationA.19A.19)g(wk) =2mXp=1p̸=kh(wk, wp) ;((A.20)(A.20)equationA.20equationA.20A.20)andG =2mXk=1g(wk) .

((A.20)(A.20)equationA.21equationA.21A.21)Then we can write A2 as the sum of the following three terms. The first comes from rzzand is independent of all wj’s and M:Azz = (−1)m2mXM=0XσM(−1)Mrzz .

((A.20)(A.20)equationA.22equationA.22A.22)The second term comes from rzl and rkk and depends only on F.AF = (−1)m2mXM=0XσM(−1)M 12ǫF . ((A.20)(A.20)equationA.23equationA.23A.23)The last term depends on h, g, and H. It isAH = (−1)m2mXM=0XσM(−1)M Xl/∈σMXp∈σMh(wl, wp) +Xk∈σMXp/∈σMh(wk, wp)+2mXk=1g(wk) −Xk∈σMXl/∈σMH(k, l).

((A.20)(A.20)equationA.24equationA.24A.24)(A.20)(A.20)equationA.25page3535

First, we will evaluate AF by demonstrating that F = 0. F is given byF =2mXk=12mYi=1i̸=kwk + wiwk −wi.

((A.20)(A.20)equationA.25equationA.25A.25)We can put all the terms in F over a common denominator to obtainF =Yi>j1wi −wjXk(−1)k2mYi=1i̸=k(wk + wi)Yi>ji,j̸=k(wi −wj). ((A.20)(A.20)equationA.26equationA.26A.26)Now we can use the fact that Q2mi>j=1;i,j̸=k(wi −wj) is the discriminant, which equals theVan der Monde determinant.

Consequently, we can write it asYi>ji,j̸=k(wi −wj) = det Mk ,((A.20)(A.20)equationA.27equationA.27A.27)where Mk is a (2m −1) by (2m −1) matrix with(Mk)ij =(wi)(j−1)for i < k ;(wi+1)(j−1)for i ≥k . ((A.20)(A.20)equationA.28equationA.28A.28)Then we can write F asF =Yi>j1wi −wjXk(−1)kYi̸=k(wk + wi)det Mk.

((A.20)(A.20)equationA.29equationA.29A.29)The expression in curly brackets looks like the determinant of the matrix A when it iscalculated by expanding the last row, where A is given byA =11. .

.1w1w2. .

.w2m............w2m−21w2m−22. .

.w2m−22mQi̸=1(w1 + wi)Qi̸=2(w2 + wi). .

.Qi̸=2m(w2m + wi). ((A.20)(A.20)equationA.30equationA.30A.30)We can write the kth entry of the last row asYi̸=1(wk + wi) = w2m−2k" 2mXi=1wi#+ w2m−4kXi

((A.20)(A.20)equationA.31equationA.31A.31)(A.20)(A.20)equationA.32page3636

From this equation we can see that the last row of A is equal to a linear combination ofthe first 2m −1 rows of A, so det A = 0. Therefore, since F ∝det A, F is zero.

Then wecan conclude that the contribution, AF , to the 2m-point function is also zero.Next, we will evaluate Azz by performing the sumS =2mXM=0XσM(−1)M . ((A.20)(A.20)equationA.32equationA.32A.32)The sum over σM is a sum over all ways to choose M objects from a set of 2m objects, soS isS =2mXM=02mM(−1)M = (1 −1)2m = 0 .

((A.20)(A.20)equationA.33equationA.33A.33)ThenAzz = −(−1)mrzzS = 0 . ((A.20)(A.20)equationA.34equationA.34A.34)Lastly, we must evaluate AH.

We will begin by proving the identityXk∈πHπ(k, l) =Xk∈πh(wl, wk) ,((A.20)(A.20)equationA.35equationA.35A.35)where π is a subset of {1, . .

., 2m} containing M elements, and l /∈π. Using the definitionsof Hπ and h, we can writeXk∈πHπ(k, l) =wlYj /∈πj̸=lwl + wjwl −wjYi∈π1(wi −wl)2Xk∈πwkYi∈πi̸=kwk + wiwk −wi(wi −wl)2 ;((A.20)(A.20)equationA.36equationA.36A.36)andXk∈πh(wl, wk) =wlYj /∈πj̸=lwl + wjwl −wjYi∈π1(wi −wl)2Xk∈πwkYi∈πi̸=k(wl + wi)(wl −wi) .

((A.20)(A.20)equationA.37equationA.37A.37)From these two equations, we can conclude thatXk∈πHπ(k, l) =Xk∈πh(wl, wk)if and only if(A.20)(A.20)equationA.39page3737

Xk∈πwkYi∈πi̸=kwk + wiwk −wi(wi −wl)2 =Xk∈πwkYi∈πi̸=k(w2l −w2i ) . ((A.20)(A.20)equationA.38equationA.38A.38)Subtracting the right-hand side from the left-hand side of this equation and simplifying,we find that the condition becomes0 ="2M−1 Yi∈πwi(wl −wi)# Xk∈π(wk −wl)M−2 Yi∈πi̸=k1wk −wi.

((A.20)(A.20)equationA.39equationA.39A.39)Because the expression in square brackets does not equal zero for arbitrary values of w,the above expression is true if and only if0 =Xk∈π(wk −wl)M−2 Yi∈πi̸=k1wk −wi= B . ((A.20)(A.20)equationA.40equationA.40A.40)To make the argument simpler, we will take π = {1, 2, .

. ., M}.

Then we can write theexpression on the right-hand side of the above equation asB =Yi,j∈πi>j1wi −wjXk∈π(−1)M+k(wk −wl)M−2 Yi>j∈πi,j̸=k(wi −wj)=Yi,j∈πi>j1wi −wjXk∈π(−1)M+k(wk −wl)M−2 det Mk ,((A.20)(A.20)equationA.41equationA.41A.41)where Mk is an (M −1) by (M −1) matrix defined as in equation (A.28). The sum in theexpression for B is just the row expansion of the following determinant:det11.

. .1w1w2.

. .wM............wM−21wM−22.

. .wM−2M(w1 −wl)M−2(w2 −wl)M−2.

. .

(wM −wl)M−2= 0 . ((A.20)(A.20)equationA.42equationA.42A.42)The last row in the matrix is a linear combination of the other rows, so B = 0.

As a result,we have shown that the identity in equation (A.35) is true.Using this identity, we can write AH asAH = (−1)m2mXM=0XσM(−1)MG +Xk∈σMXp/∈σMh(wk, wp). ((A.20)(A.20)equationA.43equationA.43A.43)(A.20)(A.20)equationA.44page3838

Because G is independent of M and σM, we can perform the sum over G using equations(A.32) and (A.33) to obtainAH = (−1)m2mXM=0XσM(−1)M Xk∈σMXp/∈σMh(wk, wp) . ((A.20)(A.20)equationA.44equationA.44A.44)When we sum over all subsets, σM, the pair (wk, wp) will take on all possible values, andeach particular (wk, wp) will occur2m−2M−1times.

ThenAH =(−1)m2m−1XM=12m −2M −1(−1)M2mXk,p=1k̸=ph(wk, wp)=(−1)m(1 −1)2m−2G ,((A.20)(A.20)equationA.45equationA.45A.45)so AH = 0 for 2m > 2. Therefore, the 2m-point functions for 2m > 2 are all zero as ǫ →0at order V 20 .The calculation for the O(V 40 ) contribution to the four-point function is similar butlonger.

There are two key differences. The first is that we must be careful to subtractout the disconnected diagrams.

The second is that we must also make use of the identityPσQi 1/(wσ(i) −wσ(i+1)) = 0, where the sum is over all permutations, σ, of 2m elements,with σ(2m + 1) = σ(1).In all the m-point function calculations we have done, the only function that we haveobtained that is SU(1,1) invariant is Qi wi/(wi −wi+1). For example, F and G are notSU(1, 1) invariant.

Additionally, the m-point functions of ˙X(ti) must be symmetric underinterchange of the ti (at least when α = 1). We find that when we symmetrize the SU(1,1)invariant function, we get 0.

This suggests that ⟨˙X(t1) . .

. ˙X(t2m)⟩= 0 to all orders in V0,but does not guarantee it.Appendix (A.20)(A.20)equationA.1appendixBB.

Ward Identity CheckIn Section 6, we noted that there is a complete set of “broken reparametrizationinvariance” Ward identities which must be satisfied by the free energy and correlationfunctions in the theory [10]. These can be derived from the condition that the string theoryboundary state to which our theory corresponds must have reparametrization invariance.This condition is implemented by requiring (2.8) to be satisfied, and in this appendix, for(A.20)(A.20)equationB.1page3939

zero magnetic field, we check the simplest form of this identity in order to show that ourrenormalization scheme is correct. It turns out that we need to know the charged sectorcontributions to the free energy and to the two-point function.

Both can be read offfromthe results in Section 6.A special case of (2.8) is(L1 −˜L−1)|B⟩= 0 ,((A.20)(A.20)equationB.1equationB.1B.1)and it is shown in [10] that this is equivalent to the following condition on the connecteddiagram generating functional W:∞Xm=−∞m̸=0(m + 1)α−m∂W∂α−m−1= il2∂2W∂X0∂α−1−∂W∂α−1∂W∂X0,((A.20)(A.20)equationB.2equationB.2B.2)where the αm are the closed string oscillators (see (2.7)), and l =√2α′. If we differentiatewith respect to αp and then set each αm = 0 we get (using the fact that the one-pointfunction∂W∂αm = 0 on any solution)∂∂X0∂2W∂αp∂α−1−∂2W∂αp∂α−1∂W∂X0αm=0= 0 .

((A.20)(A.20)equationB.3equationB.3B.3)Now∂2W∂α1∂α−1 is the first Fourier mode of ⟨˙X(τ) ˙X(0)⟩, so (B.3) with p = 1 is equivalent to(we take T = 2π)∂∂X0Z 2π0dτ2π eiτ D˙X(τ) ˙X(0)E=Z 2π0dτ2π eiτ D˙X(τ) ˙X(0)E ∂W∂X0. ((A.20)(A.20)equationB.4equationB.4B.4)This is the equation we must verify.

We now substitute the calculated forms for W (equalto TF when the sources are set to zero) and ⟨˙X(τ) ˙X(0)⟩from equations (6.7), (6.15) and(6.16) to check whether (B.4) is satisfied. The mth Fourier mode of the two-point functionfor m > 0 isZ 2π0dτ2π eimτ D˙X(τ) ˙X(0)E= m −Vr cos X0+ V 2r42 cos 2X0 δm,1 + 4 sin2 X0 −m+ · · ·((A.20)(A.20)equationB.5equationB.5B.5)(A.20)(A.20)equationB.7page4040

and for m = 1 the result isZ 2π0dτ2π eiτ D˙X(τ) ˙X(0)E= 1 −Vr cos X0 + 14V 2r + · · · . ((A.20)(A.20)equationB.6equationB.6B.6)The generating functional is given byW = −Vr cos X0 −12V 2r 14ǫ −sin2 X0+ O(V 3r ) ,((A.20)(A.20)equationB.7equationB.7B.7)and it is easy to see that (B.4) is satisfied to O(V 2r ).

Note that the contact terms playeda crucial role in this calculation, confirming that they are an essential part of the physicalcontent of the critical theory. This Ward identity represents a very non-trivial check thatour renormalization scheme is consistent.

(A.20)(A.20)equationB.8page4141

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