334 Nuclear Physacs B (Proc Suppl ) 5A (1988) 334 338 North-Holland, Amsterdam

( ' O M P U T I N G P A R T I T I O N F U N C T I O N S

Paul ( ' A R T E I / and ( ;yan B H A N O T

Superconaputer ( ' o m p u t a t m n s Research hast f lu te Florida State Unlver t i ly Tal lahassee, FL 32306

We discuss apphca t , ons of a mettaod ~e have invented recently to deteraanne the complete pa r t l tum func tmn

of fimte volunae s tat is t ical sy t l ems to ~erv good accuracy u t m g nunaerwal s inaulatmn melhod~

ha this pape r we describe a m e t h o d to compute

par ta tmn func t ions to good accuracy usang numeracal

s l m u l a t m n t echn iques A p p h c a t m n s of the me thod to

the t h ree -dmlens ,ona l Ising model 1'2 and Z ( N ) a and

SU(2) 4 gauge theor ies in four & m e n s i o n wall be sum-

marized. Us ing our m e t h o d for the Is ing model, we

were able to c o m p u t e the exponent u to an accuracy

comparab le to s t r o n g couphng expans ions from fimte

size scaling of the zeros of the p a r t n m n fune tmn 2 The

s t udy of Z ( N ) gauge theories d e m o n s t r a t e d tha t the

m e t h o d is an excellen~ way to identify weakly first order

t r a n s i t m n s a For the SF(2 ) ttaeor.~, the m e t h o d is able

to show unaanblguous ly tha t there is no phase t rans>

turn m the ze ro - t empera tu re t l aeo ry 4 T h e me thod can

be generahzed to s t u dy non-posi t ive def imte measure~ 5

(such as those at finite baryon densi ty in QCD) as well

as lo compu te the b e t a - f u n e t m n ot field theories

Let us began by describing wha t the numerwal

m e t h o d is It is easmst lo do thas in the context of

sonae &screle model . Lel us consider ltae ~hree &men

smnal I sm g model G e n e r a h z a t m n to other mode l s is

tmwal

Th e I smg model is defined as a sy s t em of sp ins on

the sites of a cubic latt ice. The energy is given by,

1 E = V ' ( 1 - s , s ; ) (1 )

4"--- ' 1~,3)

0920 5632/88/$03.50 © Elsevaer Scaence Pubhshers B V (North-Holland Physics Publashang Division)

where s = ±1, the s u m runs over n e a r e s t - n m g h b o r

pairs and we wall a s smne per lo&c boundary condi t ions

E is anteger valued and can vary from zero for the or-

dered s ta te to a numbe r Era, of lhe order of the vo lume,

for the m a x u n a l l y f rus t r a t ed slate. The par t i t ion func-

t m n as, E~

Z(u) Z P(E)~jE' (2)

E 0

where P ( E ) is the n u m b e r of s tates of the s y s t e m at

energy E, u = e 4J

Our me thod to c o m p u l e Z numen , ' a l ly it t he tel-

lowing

Dlwde up the range of E values into set', (ou ta lu -

mg four consecut ive energies each The last E value

of one set ~s the first of the next set ( 'ons lder one

of the sets To un t lahze a spin coaahguratlon oaa the

c o m p u t e r rote that energy tel . one Marts rather wi th a

comple te ly ordered or a completely d>ordered config-

u ra t ion Then randoml~ chosen spins me fl ipped, the

flip being accepled if 1he energy of the conf igura t ion is

dr iven towards one of the eaaergmsna the tel O nc e l l a e

lat t ice energy is lnmal lzed ~o be one of the eiaer~les m

ttae ,,el, tlae lattice la u p d a l e d by flipping bpms at t i les

chosen r a n d o m l y If the tpua filp keeps a lattice energ~

~attun the range of energms m the se*, ~t is accepted

These spin flap a t t e m p t s are repea ted a large imnaber

of trams a nd the n u m b e r of t imes the latt ice energy hab

P Carter, G Bhanot / Computmgpartmon functwns 335

a given E value in the sel is r ecorded Th i s exper iment

is t hen r epea t ed over all sets

The re la t ive probabi l i ty for t he sys t em to be m

one ( E ) or tile o ther ( E ' ) ene rgy s ta te in the set is

an u n b m s e d e s t i m a t o r of the re la t ive n u m b e r of s ta tes

P ( E ) , P ( E ' ) at these energy values. Because the sets

over lap m one energy value, one can also find the nor-

ina l l za tnm 111 P ( E ) between sets T h u s , the occupa-

tmn n u m b e r s In the sets gene ra te the full p a r t g l o n

funct ion , a p a r t f rom an Jrrelevant n o r m a h z a t l o n fac-

tor T h e choice of four energy values m a set is to

allow the sys ten l to reach any local spin conf igura t ion

troin ,my o t h e r conf igura t ion in one upda t e . One could

have picked inore t han four energies in a set. However,

there are p r o b l e m s wi th ergodIcl ty if one choses fewer

t han four energy values ui a set It as easy to also see

that Io keep the abso lu te er ror m the value of P ( E )

fixed, the c o m p u t e r tmle r scales wi th the volume as

T "- l "2

Using th i s m e t h o d , we have c o m p u t e d the p a r h -

t lon t unc t ion of the I s m g model on lat t ices of sizes up

tO ] 0 { N()Z It IS clear t rom Eq. 2 ttlat the p a r t m o n

fnllCt~on 1 ~, d l)olynolnlal m u wItll posi t ive coefficients

It ( an the re fo re only haxe complex zeros The s tudy

of such zero~ was first ~uggested b 3 Lee and Yang "

In llle l he r lm ,dy i l an nc lmul the z e r o s p l i lc l i itle posl

tl ' ,e u axis al the cr i t ical poin t u~ It is easy to show

tha t if one looks at t he zero ul closest 1o the real pos-

itive u axI% ItS pos i t i on scale~ with lat t ice size L as

u](L) - Ul(,X~l -- L 1/t : Table 1 shows the two zeros

closest to the pos i t ive u axis up to L - 10 o b t a i n e d us

in K our m e t h o d Fron l an analysl~ of these zeros, 2 we

found tha t l~ 0 6295{10), a Iniinl~er t ha t is as accu-

ra te as the be . t k n o w n s t rong c o u p h n g or e -expans ion

resu l t s 8

Tim ine thod desc r ibed above for the ] s t ag lnodel

T a b l e 1

2 3 4 5 0 8

10

F i r s t zero

R e t u ) h n l u )

0 202893 0 292893 0 365053 0 141742 0 384283 0 087739 0 392787(5) 0 000078(5) 0 397563(5) 0 045411(5) 0 402718(5) 0 028590(5) 0 405405(5) 0 010900(5)

S e c o n d ze ro

R e ( u ) h a ( u )

0 1 0 29311 0 23061 0 34440 0 14330 0 30570(5) 0 00833(5) 0 37757(5) 0 07277(5) 0 30011(5) 0 04535(5) 0 30052(5) 0 03172(5)

The two zeros of the parl~tTon functzon cloa- est to the ~nfinzt~ volume cT~hcal point uc as a func twn of lattice ~: t L The results shown wffhout errors arc ~ xact, the part~twn funclzon ~.s known cxactlg

d e a r l y generahzes to any tl leory where ttae ac t ion takes

a d iscre te set of values Let us consider next the Z(21

gauge theory" in four d imens ions The var iables are

again eteinents of Z ( 2 ) and defined on the l inks of the

]Mtlce T h e energy, is ~lven by.

1 E ( 1 - (,ssss)r I (3) E 2

P

where ( s s s s ) p deno te s the p roduc t of hnk var iab les on

a uni t p l aque t t e and the suln runs over all the plaque-

t ies P T h e pa r t i t i on func t ion is given by Eq 2 except

tha t u ~ .,3 One know~ f rom duality a n d nunlerl-

cal Sl lnulat lons ~ tha t t he theory has a f i r s t -o rder phase

t r a n s i t i o n al @ Ln(1 r "¢~)

One can d e t e r m i n e the par t i t ion fui lct lon of this

theory USUlg lhe t eeh iuque descr ibed for the Isln~

mode l Figure 1 show~ t h e order p a r a m e t e r ,g = -

I s ss . s )p "- ob ta ined f rom tile pa r t i t i on funct ion mea-

s t l reinent ~ The phase t r a n s i t i o n at 3~ s t ands out

clearly in Figure 1 If one looks al the zeros of t he

p a r t i t i o n tunct lon for v i n o u s ~olumes. one finds the

r e su l t s shown in F igure '2 T h e ra te at winch these ze-

r o ~ a p p r o a c h the posit ive u axis determllmSl~ For t he

d a t a of F igure 2, one finds 1_ 4 06(10) This is pre- i,

clsely wha t one expects for first o lde r t r ans i t ions Tile

336 P Carter, G Bhanot / Computmg partttton Jumtwns

i. ..... ............. /_

0 n 5

r? b

1

r

~ n 2

,t i~.o 1.5

Ftgure 1 The order p a r a m O e r < S > - < ( s s s s ) p

plot ted as a f u n e t w n of A on var ious latt ice s~:(,~ fo~

tht Z(2) gauge theorg ~r~ f our & m e n s w n s The i,er-

hcal hne zs the location of the ~nfinzte l, o lume phase

t r a n s O w n poznt The lattice szze ~ncreases f rom L -

2 to L - 6 clockwise f~om bo t tom left

0.05

0.0~

0.02

0.01 i

0 30 L , 015

' ' ' ' I '~'

' i .... 1

!

I, 1

t

!! @66 II 1- 0,18

Re(u )

F~guT( 2 Zeros ~n the complex u e ~'~ plane nea~

the pos~tu'e u ax~s f o r the Z(2) gauge theo~.q on 34,44

and 5 4 lattices No te 1he a c c u m u l a t w n of zeros to-

ward~ the p o ~ h t , e u az~s and their approach (as tht

lathce ~ze increase. , ) tou ,aM, ~ 0 171573 , th~

emtwu l point o f tht th¢orq

po,ilt to no te is tha t 11 is poss ib le to ~et a good mea -

burexneni of u r a t h e r easdy u s ing our m e t h o d Indeed

for f i r s t -o rder t r a n s l t m n s , w h e r e the co r re lauon l eng th

lS t ] I l l t e a t t h e t r a l l q l t l O l l po in t , the l a t t i c e - l z e s Ol le

would have to s t u d y would also be smal l This Illeans

tha t Jt would be easy m our m e l h o d to use the scahng

of the zeros to d i s t l n g m s h a weakly f i r s t -order t runk>

tlOll where t) - I / d , f rom a ~eeond-order t r a n s l t m n

O u r m e t h o d also Kenerahzes to theor ies where the

enerKy takes oil con tmuou~ values m a b o u n d e d do-

nlaln For such a sys t em, one would d l w d e the energ~

range into a l a rge n u m b e r of ln iervals , inake sets of in-

tervals as before and es t imate , i n s t e ad of the cont inu-

ous t u n c t l o n P ( E ) t h e a p p r o x i m a t e f u n c t m n P ( E , )bE,

at the mldpo ln t~ of the lnlerval~

\¥e }lave s t u d m d m tlu~ way the lat t ice gatlKe the-

or 3 11l four dllnell%lOll ~, F1p~ure "~ shoves the zeros o]

the Wi l son ,flU(2) la t t ice gauge theor ) TM on a 2 4 lat-

tice We have p lo t t ed all the zeros ob t a ined wi th an

Wtlson Action

O 5 - • 7

t_ 001 . . . . . . .

. . . . I O5

Re(u

:o

• - , . ]

I C1 5 1

F~gure t The ze 7o, on a 24 lal&ce f m the I I ' d so .

theory The vurmble p lo t ted ~, u ~ ,~14 3o l ,

that ther¢ ~s no hne of zeros p inch ing th~ ~eal u aa*~

~xcepl ~l~ar u 0 (fl - .~.) eo r re spandm9 to th~

a ,gmp to t t ca l lg free f ixed point o f the theory A'ot,

also the prcacnc( o f a s h o u h h l of z( ro,~ near d 2 It

(o~Tesptmdtng to lhe cros,sore~

a rb l t r a rx pal ta t lon of the energy lan~e T h e m u n b e r

of zeros in the t rue t h e o r y lllay be llllleh sma l l e r (or

lar~er) t h a n t ins TIu~ 1, a pou~t we haxe not yel s tud

P Carter, G Bhanot / Computmgpartmon functions 337

rod. What Js clear f rom Figure 3 however is tha t there

is an excluded region around the real u = e -~/4 axis

where there are no zeros This Is a direct demonst ra -

tion that there is no phase transi t ion in the theory for

finite values of '3 In addi t ion, one can see a shoulder

near the posltlon of the well known crossover m this

theory from weak coupling (continuum) behavior and

to coupling behavior It is also evident tha t there is

a line of zeros accunmlat lng towards u ~- 0, which is

indeed the place where the theory has an asympotot i -

call 3 free fixed point hi summary, all the features of

the gauge theory that have been revealed by extensive

lat t ice calculations are visible at a glance in Figure 3

It is obvmus 4 that if one obtains the posit ion of the

zero closest to u - 0 on lattices of size L and L ~ (L and

L r largel, then these are related to the beta-funct ion

B(g 2 ) by,

~ 9~, dg2 = Ln L

. t B(g2) (4)

where q" - ~ Thlq calculation of measuimg the beta-

function of non-Abelaan theories from finite size scaling

of llq ze~ob has not yet t)een carried (nil

Om method can also be apphed m si tuations

where nornlal Markov type sunulatmn techmques do

not apply For exalnple, if one studies QCD at finite

baryon density (SU(3) gauge theory with a non-zero

chenncal potential) , one finds tha t after integrating out

the fermlons, one is left with a de ter rmnant which is not

posit ive-definite Tins means that normal simulation

techniques, which rely on a probabdls t ic in terpre ta t ion

of the par t l tmn functmn, do not apply. However, the

teclnnque we use, which inerely amounts to determin-

ing, tlle weight of configurations with a given energy in

phase sp,~ce, can still be used effectively The reader is

directed to Ref 5 for some fur ther details

To summarize, I have descr ibed a numerical meth-

od to s tudy statistical systems which shows consider-

able promise. It can be used to compute critical expo-

nents to high accuracy, to dist inguish weakly first-order

t ransi t ions , compute the beta- funct ion of theories as

well as to simulate theories where the plobabahty in-

t e rpre ta t ion fails

Acknowledgement :

This work was suppor ted by the Florida State Um-

versity Supercompute r Computa t ions Research Insti-

tute which is partially funded by the U S. Depart-

ment of Energy through contract number DE-FC05-

85ER250000

R E F E R E N C E S

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