Nuclear Physics B (Proc. Suppl.) 17 (1990) 691-693North-Holland

CRITICAL BEHAVIOUR OF NON-COMPACT LATTICE QED WITH A FOUR-FERMION INTERACTION

Stephen BOOTH, Richard KENWAY and Brian PENDLETON

Physics Department, Edinburgh University, Edinburgh EH9 3JZ, Scotland

Alan HOROWITZ

Universitât Kaiserslautern, Fachbereich Physik, D6750 Kaiserslautern, Germany

Numerical results for the chiral-symmetry-breaking phase transition in non-compact QED with a four-fermioninteraction, obtained using staggered fermions and an 84 lattice, are compared with the solution of the gapequation for the pure four-fermion model on the !5.ame size lattice. Agreement between the two suggests thatthere is no evidence for non-mean-field critical behaviour in the numerical data .

1. INTRODUCTIONIn this article we present numerical results for the

chiral-symmetry-breaking transition in a lattice modelof four-dimensional QED with an additional chiralsymmetric four-fermion interaction 1. This work wasmotivated by two recent results.

One is that non-compact lattice QEDwith masslessstaggered fermions, whose lattice action is

2 0,v>l`

+ F X(x)D(8).yx(y)N'y

7I0(x) = (-1r°+ . . .+m"_ J

where 9. takes values in the real line, appears to have asecond order transition at strong coupling 2 . Claimshave been made that for small numbers of fermionflavours (less than about four) this transition may havenon-mean-field critical exponents 2, 3. The evidencefor non-mean-field behaviour in the data is based onpolynomial extrapolation of the data for (XX) at non-zero fermion mass on a finite lattice to zero fermionmass . This procedure is suspect in the critical regionbecause of the significant fermion-mass dependence ofthe data 1, 3 and the intrinsic problem that the resultof the extrapolation should be strictly zero . In spite of

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691

tnese reservations, the extrapolated data for zero, twoand four flavours appears to be consistent with the crit-ical behaviour predicted by an approximate solution 4. 5of the Schwinger-Dyson equation for the fermion self.energy in quenched QED.

The second result comes from including a chiral-symmetric four-fermion interaction in the approxi-mate analysis of the Schwinger-Dyson equation forQED 5, 6. 7 , which predicts a critical line in the planeof the electric charge and four-fermion coupling alongwhich the anomalous dimension of ~O is large . Thisphase diagram is supported by our numerical simula-tions for four flavours 1 .

2. SIMULATIONSOur simulations are for the system obtained by in-

cluding the interaction

- G1: z(x)x(x)z(x + A)x(x +A)3 .96

in eq.(1 .1). Configurations on an 84 lattice were gener-ated by the Hybrid Monte Carlo algorithm 8, with trajec-tories of unit length in molecular-dynamics time. Ourfermionic boundary conditions were periodic in spaceand anti-periodic in time . Simulation details have beenpublished elsewhere 1.

In figure 1 we show the results for (Xx) at a fermion

mass of 0.05. We have fitted a smooth surface through

the points, without taking into account the errors . The

figure indicates that the chiral-symmetry breaking tran-

692

S. Booth et al./Critical behaviour of non-compact lattice QED

0.4

0.3

0.7-

0.1

ao

Figure 1: (zX) at a fermion mass of 0.05 for differentvalues of Q and G.

sition persists for G > 0. There appears to be a line oftransitions in the ß-G plane connecting the "on-axis"transitions found previously 1, 9.

We have tested the conjecture that there is evidencefor non-mean-field critical behaviour in our numericaldata . The data available for analysis is for a fixed lat-tice size (84), at non-zero fermion masses, at two sec-tions approximately transverse to the critical line: onecoincides with pure non-compact QED, the other to afour-fermion theory with relatively weak gauge coupling(where large anomalous dimensions are predicted by theapproximate Schwinger-Dyson analysis y) . Because ouranalysis is at fixed lattice size, it is necessarily crude andwe cannot reliably extract predictions for critical expo-nents. We reject any extrapolation of the data and,instead, study the dependence of the chiral condensateon fermion mass and couplings close to criticality.

We compare our results for (XX) with the solutionof the gap equation for the pure four-fermion system,with lattice action

=

E [I2

n~(z)X(z)~X(a+A) - X(x - WAm

+mgX(z)X(x)]

-Gg7C(a)X(~)X(a+W)X(z+ A) .

(2.2)046

The gap equation for this system on an L4 lattice

is 9.10

1

mg+8Gg(XX)g(XX)g = L4

(mg+ 8Gg(XX)g)Z + E, sin'p.(2.3)

where

z=e,P,

orz*(ri

"+'~ , (n,, = 0, . . . , L -1), de-= L

Lpending on whether the fermionic boundary conditionsin the P direction are periodic, or anti-periodic, respec-tively.

The solution of the gap equation exhibits mean-fieldcritical exponents 9. 10 . We regard agreement betweenthe data and the gap equation as indicating that thedata contains no evidence of non-mean-field critical be-haviour.

In fitting the solution of eq.(2.3) to the numericaldata we allow for the following four free parameters :

(2 .4)

Gg

=

c'GG+ Co

(2.5)

Mg

=

cnm.

(2.6)

(XX)g = CRX(XX)

The freedom to vary these parameters does not alterthe mean-field nature of eq.(2.3). What is at issue ishow accurately eq.(2.3) fits the data in what appears tobe the critical region .

'the analysis of our data for non-compact QED(with four flavours) in the absence of four-fermion in-teractions is presented in an accompanying article 11 .

Here we present the analysis for the four-fermion modelat fixed gauge coupling of Q = 2.0 . This is the re-gion of couplings for which the approximate solutionof the Schwinger-Dyson equation for the fermion self-energy predicts a large anomalous dimension for ~0 .Preliminary data were presented previously 1; we haveexpanded this data set in what appears to be the criticalregion 12 . In all cases, 100 trajectories were discardedfor equilibration, and measurements were made over aminimum of a further 200 trajectories. The data isplotted in figure 2, along with our best 2-parameter fitto the gap equation with parameter values:

Gg = 0.63G +0.04

(2.7)

We found it unnecessary to rescale the fermion massm and the condensate (XX) in order try obtain a good

0.3

0.2

0.0

Figure 2: Data for (XX) versus G for the four-fermion

theory coupled to non-compact QED at Q = 2.0 onan 84 lattice at mass values m = 0.1, 0.05, 0.025and 0.0125 (from top to bottom). Superimposed is

our best 2-parameter fit to the gap equation of a purefour-fermion model.

fit . The non-zero value of co may be explained from thefact that G = 0 is still an interacting theory, due to theeffective electromagnetic four-fermion interaction . The

same effect is presumably responsible for the rescalingof G.

3. CONCLUSIONSOur results indicate that the phase transition in non-

compact QED with 4 flavours including the effects of

a chirally invariant four-fermi interaction is consistent

with being a second order transition with mean-fieldcritical exponents.

ACKNOWLEDGEMENTSWe wish to thank Peter Hasenfratz, Martin Lûscher,

Tony Kennedy, Rainer Sommer and David Wallace for

useful discussions . This work was supported by SERC

grant GR/E 8696.3 . The simulations were carried out

on the the Edinburgh Concurrent Supercomputer, the

major support for which comes from the SERC through

grant GR/E 21810, the Computer Board, the Depart-

S. Booth et al./Critical behaviour of non-compact lattice QED

0.1 0.15 0.2 0.25 0.3

G

693

ment of Trade and Industry and Meiko Ltd. SPB ac-knowledges support from Meiko Ltd. AMH is sup-ported by Deutsch Forschungsgemeinschaft grant # ME567/1-2. BJP is supported by the SERC.

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