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FERMIONIC EFFECTIVE ACTION AND THE PHASE STRUCTURE

arXiv:hep-lat/9305017v1 20 May 1993DFTUZ 93.08, May 1993FERMIONIC EFFECTIVE ACTION AND THE PHASE STRUCTUREOF NON COMPACT QUANTUM ELECTRODYNAMICS IN 2+1 DIMENSIONSV. Azcoiti, X.Q.

Luo and C.E. PiedrafitaDepartamento de F´ısica Te´orica, Facultad de Ciencias, Universidad de Zaragoza,50009 Zaragoza (Spain)G. Di Carlo and A.F.

GrilloIstituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati,P.O.B. 13 - Frascati (Italy).A.

GalanteL.N.F - I.N.F.N. and Dipartimento di Fisica, Universita’ dell’ Aquila, L’ Aquila 67100, (Italy)ABSTRACTWe study the phase diagram of non compact QED3 using themicrocanonical fermionic average method described elsewhere.

We presentevidence for a continuous phase transition line in the β, N plane, extendingdown to arbitrarily small flavour number N.1

In this Letter we continue a systematic study of Lattice Abelian modelswith dynamical fermions [1,2,3] that we are making through the use ofthe microcanonical average method [1] for dealing with fermionic latticesimulations.Earlier work was related to four dimensional compact [1], and noncompact [2,3] models.In particular the non compact abelian model isinteresting, being a candidate for a strongly interacting continuum theory:in this case we have presented detailed analysis of the phase structure, bothin (β, m), including m = 0 [2] and (β, N) [3] planes.Lower dimensional models have a particular interest on theoreticalgrounds. In 1 + 1 dimensions, quantum electrodynamics at zero mass ( i.e.the Schwinger model) is analytically solvable, confining and asymptoticallyfree; in this case lattice results can be directly related to exact (continuum)ones.The 2 + 1 dimensional Abelian model shares some features withSchwinger’s:it confines static charges and is asymptotically free, soits understanding should be relevant to more physical theories in fourdimensions.Although not solvable, the model is super renormalizable.It is also potentially interesting in relation to models of high Tcsuperconductivity [4].In this paper we describe our study of the phase structure of noncompact QED3 in the (β, N) plane.

We find a critical line that consists oftwo segments, a first order line in the large-N and a second order one in thesmall-N region. Our analysis is based on the studies of two quantities, theeffective fermionic action and the chiral condensate.

An intriguing featurethat emerges from these studies is that the phase diagrams obtained usingthe two quantities differ in the low-N region. The analysis of the fermionicaction suggests that the second order critical line terminates on the N = 0axis at finite coupling, βc = 0.49, implying the existence of two phases evenin the quenched theory.

The behavior of the chiral condensate, however,shows no restoration of chiral symmetry for any finite β, in the investigatedrange, implying that the nonanaliticities occuring at βc = 0.49 are notrelated to the chiral symmetry breaking, in agreement with the conclusionof previous studies of QED3 [5,6,7] as well as with theoretical expectations.We mention that extensive amount of work on the subject has beendone in the continuum formulation of the model. This has been reportedin refs.[8-11].

Also, simulations of the compact variant of QED3, with theuse of the method described below, have been presented in [12].The method we use, is based on the introduction of an effectivefermionic action SFeff(E, N, m) [1-3]. An advantage of this method is that itallows simulations to be performed exactly in the chiral limit.

The effectivefermionic action, which is a function of the pure gauge energy E, fermionmass m and number of flavours N, is defined through,2

e−SFeff (E,m,N) =R[dAµ(x)](det ∆(m, Aµ(x)))N/2δ( 12Px,µ<ν F 2µν(x) −3V E)R[dAµ(x)]δ( 12Px,µ<ν F 2µν(x) −3V E)(1)where ∆(m, Aµ(x)) is the fermionic matrix (we use 4-components staggeredfermions) and E is the normalized pure gauge energy. The denominator in(1) is the density of states at fixed energyN(E) = CGEV −32(2)with CG being an unimportant (divergent) constant and V the latticevolume.After the definition of the fermionic effective action, the partitionfunction of this model can be written as a one-dimensional integralZ =ZdEN(E)e−3βV E−SFeff (E,N,m)(3)from which we can define an effective full action per unit volume as¯Seff(E, β, N, m) = −ln E + 3βE + ¯SFeff(E, N, m)(4)¯SFeff(E, N, m) in (4) is the fermionic effective action (1) normalized to thelattice volume.Once the effective action, which entirely characterizes the fermioniccontribution, is defined, the qualitative features of the phase structure ofthe system can be studied analytically.

The integrand in (3) is a stronglypeaked function of E. Since the effective full action diverges lineary withthe lattice volume V , one can evaluate the free energy and its derivativesby saddle point. The mean plaquette energy < Ep >= E0(m, β, N) will begiven by the solution of the saddle point equation [2]13E −β −13∂∂E¯SFeff(E, N, m) = 0(5)satisfying the minimum condition1E2 + ∂2∂E2 ¯SFeff(E, N, m) > 0(6)By differentiating equation (5) respect to β we get the specific heatCβ = ∂∂β < Ep >= −{13E20(m, β, N) + 13∂2∂E2 ¯SFeff(E, N, m)E0(m,β,N)}−1(7)3

The effective action must be continuous even in the thermodynamicallimit; however its derivatives may be discontinuous in this limit.Adiscontinuity in the first energy derivative implies an analogous behaviourfor the average energy, hence a first order phase transition.If thefirst energy derivative is continuous but some higher order derivative isdiscontinuous, then the average energy will be continuous, the specific heator some of its derivatives being discontinuous, i.e. the system will undergoa continuous phase transition [2].This is not the only way in which a phase transition is generated,however, since this can be obtained also through a cancellation of the twoterms of the denominator in (7).

In order that this happen, the secondenergy derivative of the effective action should be negative. Then, for arange of energies and N large enough, the denominator will be negativeand there will be no solution of the saddle point equation, i.e.

there willbe a range of energies not accessible to the system, indicating again a firstorder transition.Finally, if the effective action is non analytic in the flavour number Nor the fermion mass m, this will cause again phase transitions in the (β, N),(β, m) planes.To summarize, the phase structure of the theory can be entirelydescribed in terms of the behaviour of the effective action as a function ofthe energy and the bare parameters. If it is non analytic in the energy, thena phase transition will appear.

On the other hand, if the effective action isanalytic in E, N and m, then the only other mechanism for producing aphase transition, if any, will generate a first order transition line ending ina second order point. Obviously these mechanisms can coexist, as in QED4[3].We stress that this characterization of the phase structure of the theory(more transparent in the non compact model since the density of states isknown analytically), is rather independent on the details of the numericalevaluation of the effective action.

Also, it does not depend on the evaluationand extrapolation of the chiral condensate, which, as for past experience,is very delicate especially in small lattices and in the three dimensionaltheories [6]. In fact, the above characterization of the phase structure isentirely independent from those existing in the literature for these models.The phase structure of the four-dimensional theory, obtained in this way[2] is in very good agreement with the one obtained in [13].We now present our results for the effective action.

For the fermioniceffective action we can write down an expansion in cumulants [2],−SFeff(E, N, m) = N2 < ln det ∆(m, Aµ(x)) >E+N 28 {< (ln det ∆)2 >E −< ln det ∆>2E} + ...(8)4

where < O >E means the mean value of the operator O(Aµ(x)) computedwith the probability distribution [dAµ(x)]δ( 12Px,µ<ν F 2µν(x)−3V E)/N(E).Expression (8) is a N expansion of the fermionic effective action.Following the general method described in [1,2], we have donesimulations in 63, 103, 143, 183 and 203 lattices.In Fig.1 we presentthe Fermionic Effective Action for two massless flavours on an 143 lattice,computed using the first two contributions to the cumulant expansion. Thethird one has been found compatible with zero within errors.

The resultshown in Fig. 1 is qualitatively completely analogous to the one found in thefour dimensional theory, so we can repeat the same analysis as in [2,3].

Theimportant point here is that our data strongly indicate a continuous phasetransition; in fact the Fermionic Effective Action is linear for small energiesand clearly not linear for larger energies, thus suggesting a discontinuityof the second or higher order derivatives in the thermodynamical limit atsome critical energy Ec. The first derivative is continuous, as follows fromthe analysis of the numerical data.This behaviour is dramatically evident if one plots the FermionicEffective Action minus the fit to its linear part, Fig.

2. Once the criticalenergy Ec is determined by fitting the results reported in Fig.2 with apower law function C(E −Ec)ρ, the critical coupling is computed fromsaddle point equations.

Notice that at these lattice sizes, the results fromthe saddle point aproximation are undistinguishable from the numericalones.In Table I we present the critical values of β at various values of N. Thetransition line continues down to arbitrarily small values of N, includingzero flavour (the quenched theory). The critical values of β and E in thequenched limit, βc = 0.49(1), Ec = 0.68(1), show no variations with thelattice size, in disagreement with expectations for a phase transition withdivergent correlation length, and might be interpreted as indicating that weare indeed observing a transition with finite correlation length.

Howeverthe similarities of these results with those obtained in the four-dimensionalnon compact model [2,3], cast doubts on the previous interpretation. Infact our determination of the critical couplings βc in the four-dimensionalmodel with two and four dynamical flavours, were in very good agreementwith those reported in [13], the last obtained in much larger lattices.

Butthere is no doubt that the correlation length diverges at the critical pointof noncompact QED in four dimensions.Concerning the phase structure at large number of flavours, here thediscussion in [3] also applies: at the critical values N = 6.10, Ec = 1.58 thedenominator in (6) is zero and for larger values of N it becomes negative insome energy interval. In such an energy interval, the saddle point equationhas no solution, producing a first order transition line.

As described in[3] the continuous transition line merges into the first order one, since theenergy at which the effective action becomes non analytic falls into the5

energy interval not accessible to the system, which widens with the energy.Notice that the critical energy obtained from our simulations seems to beindependent on the flavour number at small N, as in the four dimensionalcase [2,3]. In Fig.

3 we present the complete (β, N) phase diagram of themodel, at m = 0. This phase diagram has been obtained using only thefirst term in the expansion (8).The only results on the phase structure for the three-dimensional noncompact case are those in [5,6,7], suggesting a continuous, chiral symmetryrestoring transition ending at N ≃3 −4, β = ∞.On the other hand,the results reported in [6] show unambiguously that chiral symmetry isspontaneously broken in the quenched model for β values larger than ourβc = 0.49, thus suggesting that the phase transition we observe does notrestore chiral symmetry, contrary to what happens in the four dimensionalnon compact model.

This is not surprising since quenched QED3 confinesstatic charges for any finite β and there are general arguments suggestingthat confining forces make the chiral symmetric vacuum unstable [14]. Infact our numerical results for the chiral condensate in the quenched modelsupport well this scenario.In Fig.4 we plot the inverse logarithm ofthe chiral condensate against the inverse logarithm of the fermion massfor several values of β in the 143 and 183 lattices.

This kind of plot wasproposed in [13] as a very efficient way to get the critical β and the valueof the δ exponent in a continuous chiral restoring transition. The resultsof Fig.

4 show unambiguously that the ”critical β” obtained in this waymoves significantly towards larger values when the lattice size changes from143 to 183, indicating that the observation of a ”critical β” (which in thisplot corresponds to a straight line which passes through the origin) is apure finite size effect, the critical coupling being pushed towards ∞in thethermodynamical limit.We try here to discuss on the reliability of our results concerning thecontinuous phase transition line at small N.A first criticism might bethat the method we use forces for some reason a phase transition througha (non physical) non analiticity of the effective action.We argue thatthis is extremely unlikely: on one hand, our determination of the phasediagram of QED4 through the Effective Action is in very good agreementwith those obtained using traditional methods (using the behaviour of thechiral condensate); this agreement extends to all the physical observablesmeasured [2]. On the other hand we have obtained within this approachpreliminary results for massless QED2 (the Schwinger model), showing agood analytical behaviour of Seff, i.e.

the absence of phase transitions atfinite β in the one flavour model, with a scaling behaviour in agreementwith simple dimensional counting.In Fig. 5 we plot the mean value of the normalized singular part ofthe fermionic action in the quenched limit against β in a 183 lattice.

Thissingular part is defined as6

SsingF(β) = 1V < logdet∆> −a0 −a13β(9)where a0 = 0.145, a1 = −0.256 are the zero energy intercept and the slopein the small energy region of the first cumulant contribution to the fermioniceffective action (8). The existence of two phases is evident in this figure.From a numerical point of view, we must note that our critical coupling atN = 0, βc = 0.49, corresponds to a region where the results for the chiralcondensate reported in [5,6] show a very rapid change.

Unfortunately this βregion has not been explored intensively in [5,6], so no definite conclusionscan be extracted from their results.The phase structure of the model in the (β, N) plane (Fig. 3) shows theexistence of two completely separated phases.

However, chiral symmetryshould be spontaneously broken in both phases since it is broken forsmall N including the quenched limit.We have also explored if thecontinuous phase transition line is related to the percolation of topologicalstructures associated to the lattice regularization. However, the monopole[15] and string densities at the critical values of β are too small toproduce percolation of these objects.

Therefore we have at the moment nocompelling evidence for this interpretation. We want to notice here that thiscontinuous transition was not observed in the simulations of the compactversion of this model [12] thus suggesting again important qualitativedifferences between the compact and non-compact regularizations, like inthe four dimensional case.Conversely the first order line is clearly produced by pure fermioniceffects, like in the four dimensional non compact model.

We would liketo remark that our quantitative results for large N could change whenincluding all the cumulants in the expansion of the effective action. Whatcan be analytically proved is that the effective action is linear with N in thelarge N limit [3] so higher order terms in the cumulant expansion conspirebetween them in order to give this linear behaviour.

However the mainqualitative features like the fact that the first order line does not interceptthe β = 0 axis [3] remain unchanged.Concerning the possibility to have a chiral restoring phase transition,we have not seen evidence for such a transition.However we can notexclude a non analiticity of the effective fermionic action as a functionof N which could originate this transition. Indeed our approach is basedin an expansion of the fermionic effective action in powers of N and theimplicit assumption that the convergence radius of this expansion is ∞.Several interesting issues emerging out of this study and which areleft open at this moment are:the physical origin of the continuoustransition, qualitative differences between the strong and weak couplingphases, correlation length and critical exponents associated with thecontinuous transition, possibility to define a non superrenormalizable, but7

renormalizable field theory, etc.. A more detailed study of these issues isunderway.The numerical simulations quoted above have been done using theTransputer Networks of the Theoretical Group of the Frascati NationalLaboratories, of the University of L’ Aquila and the ReconfigurableTransputer Network (RTN), a 64 Transputers array of the University ofZaragoza.We thank A. Kocic for extremely interesting discussions and for acritical reading of the manuscript.This work has been partly supported through a CICYT (Spain) - INFN(Italy) collaboration.8

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FIGURE CAPTIONS1) Normalized fermionic effective action as a function of the pure gaugeenergy on a 143 lattice, m = 0.0 and N = 2.2)The order parameter, obtained from the results of Fig. 13)Phase diagram on the (β, N) plane.4)Inverse logarithm of the chiral condensate against the inverse fermionmass logarithm for several values of β in the 143 (4a) and 183 (4b)lattices (quenched case).5)Singular part of the mean value of the effective action normalized bythe lattice volume V against β in a 183 lattice (quenched case).11

TABLE CAPTIONI)Critical values of β at several values of N on the 183 lattice.12

0.51.01.5EFigure 1-0.2-0.100.1S_effF

0.51.01.5EFigure 200.020.040.060.080.10Seffsing

0.51.01.5βFigure 500.020.040.060.080.10SFsing

nfctransition00.0continuous0.continuous0.continuous0.continuous0.continuous0.0continuous0.0,0.discontinuous,continuous0.0,0.0discontinuous,continuous00.0discontinuousTableI

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