LNF 92/110 P December 1992

arXiv:hep-lat/9212020v1 17 Dec 1992LNF 92/110 P December 1992LARGE N LATTICE QED.V. AzcoitiDepartamento 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).ABSTRACTWe study the β, N critical behaviour of non compact QED with Nspecies of light fermions, using a method we have proposed for unquenchedsimulations.We find that there exist two phase transition lines:one,second order, and the other, first order, that approaches asymptoticallythe β = 0 axis.

These two lines have different physical origin, the secondone being entirely due to fermions effects.We discuss the effect of theapproximation used, in terms of an expansion of the effective action inpowers of N, and conclude that the general features should not be affectedby this approximation.1

Strongly coupled lattice QED has been, in the last years, the object ofmuch attention, both analytical and numerical. The main subject of theseinvestigations has been the question if a strongly coupled abelian modelcan be constructed with non trivial interactions in the continuum limit.During some time, the results about the nature of the fixed point ofthis model were controversial in the sense that different groups reporteddifferent values of critical exponents respectively consistent or inconsistentwith a mean field description of this model.

More recently, an extensiveinvestigation of the quenched model in large lattices and at small fermionmasses [1,2] gave evidence that a precise determination of the criticalcoupling βc is needed in order to extract critical indices with good accuracy,since small variations in the value of βc may induce strong changes in thevalues of the critical exponents towards their mean field values [3].The results of [1,2] reporting non mean field values for the criticalexponents in the quenched model, have been improved for the unquenchedcase in [4,5] and also by us in [3,6] in a completely independent calculationwhich allowed the numerical determination of βc from the results of theplaquette energy without the need of any kind of extrapolation at zerofermion mass. Additionally, an analysis, based on Renormalization Groupapproach, performed in [6] on the results for the fermionic effective action,pointed again to a non gaussian nature for the strongly coupled fixed pointof this model.

These results are in contrast with those in [7].The agreement between the results of refs. [4,5] and those of refs.

[3,6]for the two and four flavour cases was certainly encouraging and suggestedthe existence of a non trivial continuum limit for strongly coupled QED, atleast if the number of dynamical flavours is less or equal than four.Another intriguing result reported in [5] was that critical indices forthe two and four flavours models, obtained from numerical simulations andthe use of the equation of state, were found compatible between them andalso with the critical exponents of the monopole percolation transition, thussuggesting that different flavour models are in the same universality class.This result can be also understood in our approach, if the criticalvalue of the plaquette energy is independent on the number of flavours, assuggested by the results of [3]. In such a case, the value (for instance) ofthe δ exponent which measures the response of the system to an externalsymmetry breaking field will be given by the dominant contribution to theexpansion of the chiral condensate in powers of the number of flavours Nat fixed pure gauge energy Ec [3], and it should be independent of N.The aim of this letter is to analyse in detail the dependence on thenumber of flavours N of the physical results and in particular to investigatethe phase diagramm of this model in the N, β plane with special attention tothe analysis of the physical origin of the different phase transitions observed.The approach we use to simulate noncompact QED with dynamicalfermions is that of refs.

[3,6], originally tested in the compact model [8], and2

based on the introduction of an effective fermionic action SFeff(E, N, m)which depends on the pure gauge energy E, fermion mass m and numberof flavours N, and which is related to the gauge fields Aµ(x) by the relation[3]e−SFeff (E,m) ≡E=R[dAµ(x)](det ∆(m, Aµ(x)))N4 δ( 12Px,µ<ν F 2µν(x) −6V E)R[dAµ(x)]δ( 12Px,µ<ν F 2µν(x) −6V E)(1)where ∆(m, Aµ(x)) is the fermionic matrix (we use staggered fermions),E is the normalized pure gauge energy and the denominator in (1) is thedensity of states which can be analitically computedN(E) = CGE32 (V −1)(2)CG in (2) is some irrelevant divergent constant and V is the lattice volume.After the definition of the effective fermionic action, the partitionfunction of this model can be written as a one-dimensional integralZ =ZdEN(E)e−6βV E−SFeff (E,N,m)(3)from which we can define an effective full action per unit volume as¯Seff(E, β, N, m) = −32 ln E + 6βE + ¯SFeff(E, N, m)(4)¯SFeff(E, N, m) in (4) is the effective fermionic action (1) normalized to thelattice volume.The thermodynamics of this system can be studied now by means of thesaddle point technique. The mean plaquette energy < Ep >= E0(m, β, N)will be given by the solution of the saddle point equation [3]14E −β −16∂∂E¯SFeff(E, N, m) = 0(5)satisfying the minimum condition14E2 + 16∂2∂E2 ¯SFeff(E, N, m) > 0(6)By differentiating equation (5) respect to β we get for the specific heatCβ = ∂∂β < Ep >= −{14E20(m, β, N) + 16∂2∂E2 ¯SFeff(E, N, m)E0(m,β),N}−1(7)3

The effective fermionic action ¯SFeff was computed in [3,6] as a powerexpansion on the flavour number N.The observation of two different regimes in the behaviour of ¯SFeff asa function of E, linear in the small energy region and nonlinear at largeenergies, and a carefull analysis of the numerical results allowed us thedetermination of the critical value of the energy (which turned out to beindependent on the number of flavours N) and of the critical coupling βc.Furthermore it was established in [3] that the results for ¯SFeff reported inFig.1 can be very well fitted by two polynomials with a gap in the secondenergy derivative of ¯SFeff at E = Ec which manifests itself in the specificheat Cβ as a second order phase transition.The origin of this non analyticity in the effective fermionic action is notclear at present. However, our numerical results suggest that this singularbehaviour could have the same origin as the monopole percolation transitionof the quenched model extensively analysed in [9].

In fact if we take ourvalue of the critical energy Ec = 1.016(10) (independent on the number offlavours) and consider the zero flavour limit, we get βc = 0.246(2) in verygood agreement with the critical βc obtained in [9] from the results for themonopole susceptibility in the quenched model.Going back again to expression (7) it should be noticed that a nonanalyticity of the effective fermionic action is not the only way to get adiscontinuity in the specific heat.A discontinuity in Cβ can in fact beproduced by a zero in the denominator of (7) through exact cancellation ofthe two terms in this expression [3]. To this end, a negative value of thesecond energy derivative of the effective fermionic action is necessary.

Butthis is just what happens in the large energy region as it can be deducedfrom the results for the effective fermionic action reported in Fig.1.The above arguments can be made quantitative and at the same timethe critical number of flavours Nc can be simply estimated. We start fromthe cumulant expansion of the effective fermionic action [3]−SFeff(E, N, m) = N4 < ln det ∆(m, Aµ(x)) >E+N 232 {< (ln det ∆)2 >E −< ln det ∆>2E} + ...(8)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) −6V E).

Thenumerical results for the succesive terms in the expansion (8) in a 84 latticeand massless fermions have been reported in [3], and are here extended tolarger energies. As a first approximation, we will consider only the firstcontribution to (8), which is reported in Fig.

1 for N = 4.Fitting the results for the mean logarithm of the fermionic determinantat m = 0 by two polynomials: first order for E < Ec = 1.016 and fifth for4

E > Ec we get for the second energy derivative of the effective fermionicaction∂2∂E2 ¯SFeff = 0(E ≤Ec)(10)∂2∂E2 ¯SFeff = N4 [−1.669 + 3.877E −2.494E2 + 0.494E3](Ec < E < 2.5)A simple analysis of these results tell us that in order to compensatethe pure gauge contribution to the denominator of the specific heat in (7)we need N = 13.1. If N is large but less than 13.1, the height of the peakincreases with N and just at this critical value, the specific heat diverges.Now, if we increase again the value of N, there will be an energy intervalwhere the denominator of the specific heat in (7) will be negative andtherefore no solutions of the saddle point equations (5), (6) will exist inthis energy interval.

This means that these energies will not be accessibleto the system and hence, for N > Nc a first order transition will appear.The phase diagram of massless noncompact QED in the N, β planewhich emerges from our results is plotted in Fig.2. The continuous (broken)lines represent first (second) order phase transitions respectively.Theend point of the first order phase transition line is a second order phasetransition point with a divergent specific heat.

The first order line endsat some finite β since∂2∂E2 ¯SFeff = 0 for E < Ec. On the other hand, thesecond order line merges, for large N, into the first order one since Ec fallsinto the energy interval not accessible to the system, which widens as Nincreases.Of course the quantitative results of this simple analysis could changeif we take into account higher order contributions to the effective fermionicaction (8), but the physics behind these phase transitions can be understoodat this simple level.

What is independent on approximations is the fact thatthere is no first order phase transition at β = 0 for any finite value of N.This result can be rigorously proved since the effective fermionic action¯SFeff(E, m, N) is bounded for any finite N. Therefore, when β goes to zero,the effective action (4) has only one minimum at E = ∞, whereas for anyfinite value of E the effective action is finite. Thus, at β = 0 there is nofirst order transition for any finite N.A stronger result can be proved, under the very natural assumption(corroborated by the experimental data) that the effective fermionic actionis monotonically increasing with E, namely that at N →∞< E >= 0 forall β.

This implies that, in this limit only one phase (Coulomb) does exist.In fact, since the value of the fermionic determinant is bounded, itfollows thateN ≤e−SFeff (E,m) ≤eN log ∆max(11)5

which implies that the fermionic effective action diverges linearly with N.As a consequence, for N →∞¯SeffE= 1N≈32 ln N + NK(E, m)E=0(12)and¯SeffE=E0≈NK(E0, m)(13)The monotonicity of ¯SFeff(E) then implies thatlimN→∞¯SeffE=E0 −¯SeffE= 1N= ∞∀E0 ̸= 0(14)implying that E = 0 is the absolute minimum of ¯Seff for any β ̸= 0.This, together with the previous discussion, implies that the first order linereaches asymptotically the β = 0 axis.To compare with published results on this subject, Kondo, Kikukavaand Mino [10] found, within the Schwinger-Dyson approach, a continuousphase transition line which approches the β = 0 axis asymptotically. Thegeneral discussion of the above paragraph excludes such a behaviour forlarge N.On the other hand Dagotto, Kocic and Kogut [11], in the frameworkof a numerical simulation, found evidence for a second order transition linewich becomes first order at large N, crossing the β = 0 axis (Nc ∼30at β = 0).

The probable origin of the disagreement between this resultand ours is that the critical β at N = 30 is so small that it is difficultto distinguish it from zero in a numerical simulation where metastabilitysignal can be observed also at β = 0 [11].From a physical point of view, these two phase transition lines have adifferent origin. The second order line, as pointed before, could have thesame origin as the monopole percolation transition of the pure gauge model[5,9] .

Indeed our results for the critical energy and their independenceon the flavour number N favour this interpretation. The first order lineis on the other hand produced by pure fermionic effects.When thenumber of dynamical fermions increases, the fermionic contribution to thedenominator of the specific heat becomes more and more important andhas the correct sign to cancel the pure gauge contribution.To finish let us say that these results have been also confirmed by ournumerical simulations of this model.

In these numerical simulations we havetaken into account the first contribution to the effective fermionic action(8). In our opinion, the contribution of higher powers in N is not likely tochange the qualitative behaviour depicted above.In fact, since the effective fermionic action diverges linearly with N,is bounded as E →∞and is monotonically increasing, then, barring very6

peculiar behaviours, it is convex for large E (corresponding to β →0). Insuch a case, for N large enough the two terms in (7) can always compensate,so a first order transition line will be in general present, at large N and smallβ.

In conclusion, although the numerical structure of the phase diagrammight vary, it is very likely that the qualitative features remain unchanged.This work has been partly supported through a CICYT (Spain) - INFN(Italy) collaboration.7

REFERENCES1. E. Dagotto, J.B. Kogut and A. Kocic, Phys.

Rev. D43 R1763 (1991).2.

A. Kocic, J.B. Kogut, M.P. Lombardo and K.C.

Wang, Spectroscopy,Scaling and Critical Indices in Strongly Coupled Quenched QED, ILL-(TH)-92-12; CERN-TH.6542/92 (1992).3. V. Azcoiti, G. Di Carlo and A.F.

Grillo, ”A New Approach to NonCompact Lattice QED with Light Fermions ”, DFTUZ 91.34 (1992).4. S.J.

Hands, A. Kocic, J.B. Kogut, R.L. Renken, D.K.

Sinclair and K.C.Wang, ”Spectroscopy, Equation of State and monopole percolation inlattice QED with two flavours”, CERN-TH.6609/92 (1992).5. A. Kocic, J.B. Kogut and K.C.

Wang, ” Monopole Percolation and theUniversality Class of the Chiral Transition in Four flavour NoncompactLattice QED.” ILL-TH-92-17 (1992).6. V. Azcoiti, G. Di Carlo and A.F.

Grillo, ”Renormalization Groupand Triviality in Non Compact Lattice QED with Light Fermions”,DFTUZ 91.33 (1991), to appear in Mod. Phys.

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FIGURE CAPTIONS1)First contribution to the effective fermionic action (Equation 8) in a84 lattice, m = 0.0 and N = 4. Errors are smaller than symbols.2)Phase diagram in the (β, N) plane at m = 0.0.9

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