Nuclear Physics B (Proc. Suppl.) 4 (1988) 595-599 595 North-Holland, Amsterdam

THE LATTICE GROSS-NEVEU MODEL AND THE LANGEVIN ALGORITHM

N. ATTIG*, R. LACAZE +, A. MOREL + , B. PETERSSON* and M. WOLFF*

* Fakul t~t fur Physik, Un ivers i t~ t B i e l e f e l d , D-4800 B i e l e f e l d i , F.R. Germany + Service de Physique Th#orique, CEN-Saclay, F-91191 Gi f -sur -Yvet te Cedex, France

The ana ly t i c so lu t ion o f the Gross-Neveu model on the l a t t i c e is given inc lud ing order 1/N. I t is compared wi th a high s t a t i s t i c s numerical s imulat ion using the Langevin a lgor i thm.

1. INTRODUCTION L =

The mot ivat ions for studying the Gross-Neveu

model on the l a t t i c e is both the i n t e res t in the

model per se, and the use of i t as a tes t ing

ground fo r numerical algor i thms wi th fermions on

the l a t t i c e . I t is a n o n - t r i v i a l model wi th fe r -

mions, having a d iscrete ch i ra l invar iance spon-

taneously broken for large enough number of f l a -

vours of fermi ons nf.

The model is asympto t ica l ly f ree and a n a l y t i -

ca l l y so lvable fo r large n f 1'2. For nf = ~ i t 3-5

has a f i n i t e temperature phase t r a n s i t i o n

Of course, the model also has p a r t i c u l a r fea-

tures not d i r e c t l y connected to theor ies l i ke

QCD: i t is constrained to two dimensions and

contains no gauge f i e l d s .

E a r l i e r works on the l a t t i c e aspects of the

model are l i s t e d in Refs. 6-9. The present in - 1 -

ves t iga t ion is the f i r s t to extend the ana ly t i c T c = ~ a exp(y E) ,

1 /nf expansion to the l a t t i c e regu la r i za t i on .

Furthermore, we have performed a numerical simu,

l a t i on wi th much higher s t a t i s t i c s than previous

i nves t i ga t i ons , which makes i t possible to e s t i - m~ = o ,

mate up to ( i / n f ) 2 e f fec ts . We have used the

f i r s t order Langevin a lgor i thm. In fac t , one of

the major goals of the i nves t i ga t i on is to study

the dependence o f the resu l ts o f th is a lgor i thm

on the d iscrete time step.

2. THE ANALYTIC SOLUTION

In the continuum vers ion, the Lagrangian o f

the model is

nf g2 nf

(z=l c~=l ( i )

where the ~a are n , two-component spinors. A l . n f f , - rCt ~ • •

sca lar f i e l d o(x) conjugate to ~2-1= ~ ~ Is in -

troduced I and a f t e r the formal i n teg ra t i on over

the fermion f i e l ds one obtains

o(x) 2 Sef f = nf { S - - ~ - - d2x + Tr I n ( 3 + o ( x ) ) ] , (2)

where ~ = g2 nf and the p a r t i t i o n funct ion is

Z = f [d o (x ) ] exp [ -Sef f ] (3)

I f the theory is regu lar ized e.g. by a spher ical

momentum c u t - o f f A, one can make a saddle po in t

expansion of Z, g iv ing to leading order

< o > ~ o = A e -T~/~ ( 4 )

and the c r i t i c a l temperature

(5)

where YE = .5772 is the Euler constant. A semi- 2 c lass ica l analysis gives the mass spectrum

m n = n~ s in(n~/2nf ) / (nl~J2nf) ,

m k = 2nf o / T~ , (6)

wehre m k is the k ink -an t i k ink mass.

A l a t t i c e version of the model wi th Susskind

fermions was f i r s t introduced in Ref. 6 and fu r -

ther studied in Ref. 7. The act ion is def ined to be

S = N { s ~ - ~ a + s Xx QxY x"}s ' (7) x x,y (x

0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

596 N. Attig et al. / The lattice Gross-Neveu model

- - 0 . where ×x and < are two independent N-component

Grassmann var iables per s i t e . The matr ix is

Qxy = ½ [6x,y+l + 6x,y-1] +

+ ½(- l)X1[6x,y+~ - 6x,y_~] +

+ ~ 6xy [Ox+Ox_~+Ox_~+Ox_~_~] , (8)

where x = ( x l , x2 ) and y are two points on the

two-d i~ns iona l l a t t i c e and I and 2 uni t vectors

in the resp. d i rec t ions .

As was discussed in Refs. 6, 7 in t h e l / n f e x -

pansion the continuum l i m i t of the model is the

Gross-Neveu model with

nf = 2N (9)

Note that our I = g2N = ~ / 2 . On an i n f i n i t e sym-

metric l a t t i c e , the saddle point equation is

I 1 d2p (10) l - ~ f sin2p 1+sin2p 2+~-2 ,

where ~ is a constant. The in tegra l can in fact

be performed exact ly to give

1 1 (½, ~; 1; 1 X = v~2(~2+2) F ~2(~2+2)) (11)

The strong coupling l i m i t (o large) gives

= v~ (12)

and for weak coupling

o ~ Oas = 2 ~ e -Tt/2/ (13)

containing the expected factor 2 in the exponent,

and giving also the lat t ice scale. In Figure 1

we show the exact solution (11) compared with

the results from strong coupling (12) and weak

coupling (13). In Figure 2 we show °/°as" One

sees a strong "crossover" effect followed by a

slow approach to asymptotic scaling.

We have also calculated the I/N corrections

to 5, by developing in the fluctuations around

the saddle point. The calculation is rather

lengthy and w i l l be reported in Ref. 10. We get

on an in f in i te lat t ice

G

N = ~

5 0 ~

20 ng coupling 1.0 "%_,. I° * o ~

° °l°%m e • O5

0 2

I I I 0 0 5 1 0 1S

1/A

FI GURE 1

a/Gas

1.(

0 c. I I 0 5 1

I 0 5

N = ~

I 2 I 10

I/G i 15

1/.l

FIGURE 2

I d2P 9~'(P) [ ~ + ~.(p) ] - ' (14) < o > = ~ - ~T S W E F --Ca--

where

: ( l + c o s p , ) (1+cosp2) _~ T2-~-~ - x

2 k k ~2 _ )~ sin -~ sin (-2 ~--pot )

(~= 1 x 2 k 2 k (15)

( S sin2~2~+~2 ) ( 7, sin 2 ~=1 ~=1 ( T - Pa) +°2)

For comparison with the Lanqevin data we have

N. Attig et al. / The lattice Gross-Neoeu model 597

performed the in teg ra l s (or sums on a f i n i t e

l a t t i c e ) numer ica l ly .

3. THE NUMERICAL SIMULATION

To simulate the e f f e c t i v e act ion

Sef f = N ( ~ g ~ - Tr logQ) (16) X

we used the f i r s t order Langevin a lgor i thm with

an ext ra noise for the fermion par t

o T Q-l 3Q ~n ) Ox(n+ 1) = Ox(n ) - e N(-~-- ½ 5n ' ~

+ ~ q n ' (17)

where

<~n ~n' > = <qn qn' > = 26nn' (18)

The a lgor i thm is chosen to be as s im i l a r as pos-

s ib le to the Langevin algori thms fo r QCD, be-

cause one of the main object ives was to tes t

th is a lgor i thm and the E-dependence of physical

quan t i t i es . We have therefore not used features

special to two dimensions.

The numerical s imula t ion consists of runs at

N = 6, 12, 24 and 60 wi th .08 < 1/X < 2, with

l a t t i c e s of s ize 20x20 and 40x40 , and fo r f i -

n i te temperatures also 6x60 . The data were ta-

ken fo r various values of the time step ~, with

150,000- 300,000 sweeps fo r each po in t . The ma-

t r i x invers ion ~ = Q-I~ was done by the conju-

gate gradient a lgor i thm, We demanded that the

to ta l absolute value o f the rest vector

IIQ ~ - El i < .01 (19)

Errors were c a r e f u l l y monitored by the binning

method, so that co r re la t i ons in Langevin time

were taken in to account.

In Figure 3 we show < o > for N = 60 and for

two values of X as a funct ion of the time step

e. At ~ = 0 we have also p lo t ted the values com-

ing from the ana ly t i c expression on the l a t t i c e

to order I /N. The agreement wi th the ex t rapo la -

t ion of the numerical data is qu i te good. In the

strong coupl ing region the approach to c = 0 is

G

1 761 N =60

A=4 O

• e

02'

0 26

A=07 #

I I 1 0.001 0 003

FIGURE 3

from above, fo r weak coupl ing from below. This

can be understood from a per tu rba t ion study in E

of the Langevin a lgor i thm along the l ines of

Ref. I i . Deta i ls of th is study w i l l be given in

Ref. 10.

<lal>

0 5 1 - u

05(

OZ~!

O~E O,~

O,Z~7

20x20 A=I.0

N =60 D

N =2#

N=12 x

L 0.06

I 0.12

N=6 o I

018 CN

FIGURE 4

598 N. Attig et aL / The lattice Gross-Neveu model

In Figure 4 we show < o > fo r ~ = 1 and fo r

several values of N as a func t ion o f oN. Again

the open po in t a t c = 0 is taken from an analy-

t i c c a l c u l a t i o n . Ex t rapo la t i ng our numerical da-

ta l i n e a r l y in ~ and then l i n e a r l y in I/N fo r

N = 12, 24 and 60 we get fo r N== < o > = . 5 1 1 ,

in e x c e l l e n t agreement w i th the ana l y t i c r e s u l t

fo r N = =, which is < o > = .5114. For N = 6

(n f = 12) the I/N 2 co r rec t i on which we have not

evaluated a n a l y t i c a l l y is as expected s i g n i f i -

can t ly outs ide our numerical e r ro r s .

<1~1>

06 o

OL.

02

O0 08

6 , 6 0

N : 6 0 e : 0 0001

0

0

0

0

o l I I I I 1 I

10 1.2 1/* l / A

FIGURE 5a

<1~1)

05

03

6x60

o N : 6

o ~ : 0 0005 o

+

#~o o 01 o

I I I I I I I 08 10 1.2 1~

FIGURE 5b

In Figures 5a and b is shown < Io i> on a 6 x 6 0

l a t t i c e f o r N = 60 and 6 r espec t i ve l y . Each

po in t corresponds to 300,000 sweeps. For N = 60

we see the expected behaviour corresponding to

a second order phase t r a n s i t i o n . One can e s t i -

mate 1/~ c = 1.39. For N = =, the a n a l y t i c ca l -

cu la t i on gives I / ~ c = 1.43. The d i f f e rence is

compat ib le w i th a 1 /N-cor rec t ion .

For N = 6 we get instead s t rong ly f l u c t u a t -

ing values f o r o, in disagreement w i th Ref. 9.

We are present ly i n v e s t i g a t i n g i f the e f f e c t we

see is due to k i n k - a n t i k i n k con f i gu ra t i ons , pre-

sent at f i n i t e N.

4. CONCLUSIONS

From our h i g h - s t a t i s t i c s s imu la t ion we see

tha t the behaviour o f the order parameter < o >

is in agreement w i th the a n a l y t i c 1/N expansion

on the l a t t i c e down to N = 6 (n f = 12). For the

c r i t i c a l temperature the s i t u a t i o n is more com-

p l i c a t e d . The data at N = 6 look l i k e those

expected fo r a f i r s t order t r a n s i t i o n or might

be due to K ink -an t i k i nk s ta tes .

This quest ion and the ~-dependence of the Lange-

v in a lgo r i t hm w i l l be f u r t h e r discussed in Ref.

i0 .

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Part of th is work was supported by the

Commission of the European Communities

(Twining contract n ° 86300272 FROI*PU*JUI).