hep-lat/9509056 16 Sep 1995 D PACS Numerical and phase relation and Carlo phase q in Potts expansion numbers the We recently Institut simulations ordered of at have o length the the Wolfhard t q numerically f ur Staudinger twodimensional rstorder studied Mo del phase in Physik d we the Evidence t at obtain spinspin ordered Hk Janke transition As conrmed Johannes Weg t in a strong very Correlation Abstract byproduct q phase state correlation and Cn go o d p oint correlation GutenbergUniversitat numerical agrees Stefan Potts Mainz agreement for we t Through functions nd Ha mo del with length Germany evidence Kappler  the o the with with in = energy extensive in exactly the Lengths a that q May Revised KOMA  the recent Mainz disordered d moments the ordered at known Monte large cor version

t

Introduction

Firstorder phase transitions have b een the sub ject of increasing interest in

recent years They play an imp ortant role in many elds of physics as is

witnessed by such diverse phenomena as ordinary melting the quark decon

nement transition or various stages in the evolution of the early universe

Even though there exists already a vast literature on this sub ject many

prop erties of rstorder phase transitions still remain to b e investigated in

detail Examples are nitesize scaling FSS the shap e of energy or

magnetization distributions partition function zeros etc which

are all closely interrelated An imp ortant approach to attack these problems

are computer simulations Here the available system sizes are necessarily

limited and reliable FSS analyses are of utmost imp ortance The distinguish

ing input parameter in such analyses is the correlation length which sets

the relevant length scale of the system and thus the range of validity of the

commonly employed FSS Ansatze

The wellknown paradigm to investigate these questions in detail is the q

state Potts mo del which can b e tuned from a secondorder through weakly

rstorder to a strong rstorder transition by varying the number of states

q In two dimensions many quantities of interest are known exactly and

have b een shown to have interesting algebraic interpretations in knot theory

The theoretical knowledge of correlation lengths however is still quite

limited Only in the disordered phase exactly at the rstorder transition

p oint q an explicit formula for the correlation length is known

t d t

By analogy with the Ising mo del and on the basis of previous

numerical data it has b een sp eculated that the correlation length

in the ordered phase is given by In fact very recently this

o t d t

could b e proven for a certain denition but the p ossibility of a larger cor

o

relation length was left op en In this note we rep ort a careful

o o

Monte Carlo study of the leading correlation length in the two phases which

yields unambiguous numerical evidence against this conjecture Rather our

numerical analysis strongly favors an earlier conjecture that

o t d t

Mo del and simulation

The Potts mo del is dened by the partition function

X X

E

Z e E s q

s s i

i j

fs g hij i

i

where i denote the lattice sites hij i are nearestneighbor pairs and is

s s

i j

the Kronecker delta symbol We simulated the mo del with q and

p

in the ordered phase at ln q on D lattices of size V L L

t x y

For each q we considered two dierent lattice geometries namely L L

and L L lattices with L and for q and

resp ectively To take advantage of translational invariance we used p erio dic

b oundary conditions From our exp erience with simulations in the disordered

phase we knew that lattice sizes L are large enough to suppress

d

tunneling events In fact starting from a completely ordered conguration

we never observed a tunneling event into the disordered phase thus allowing

statistically meaningful purephase measurements of energy moments and

correlation functions In Fig we show the probability distributions P e of

the energy normalized to unit area in the ordered as well as in the disordered

phase demonstrating that the two p eaks are indeed very well separated

While in our earlier study of the disordered phase we employed the

singlecluster algorithm here we found it more ecient in terms of real

time p erformance on a CrayYMP to up date the spins with a vectorized

standard heatbath algorithm With this up date algorithm the value of the

spins in the largest spanning cluster never changed as would clearly b e the

case with xed b oundary conditions The main dierence b etween the two

types of b oundary conditions is that with xed b oundary spins excitations

of disordered bubbles would b e rep elled from the walls while with p erio dic

b oundary conditions they can freely move around the torus The statistic

parameters of the runs are collected in Table where we give the number of

up date sweeps in units of the integrated auto correlation time of the energy

which is roughly the same for the two lattice geometries

int e

To determine the correlation length we followed Gupta and Irback

o

n

and considered the k nL momentum pro jections i i i

y x y

y

X

n

n ik i j

y y

y

g i j ie h

x x s s

i j

L q

y

i j

y y

For n the pro jections are free of constant background terms and similar

to our analysis in the disordered phase we tried to determine from ts of

o

n n

g x g i to the Ansatz

x

L x L x

x x

n

b chc g x a ch

n n

o o

q

n

with n L Tests for the Ising mo del suggested that

o o y

o

this is a go o d approximation for Ln For the Potts mo dels with

L we exp ect that is smaller than by ab out for the k

o y

o

pro jections and for the k pro jection on the L L lattice resp ectively

x

Even though in the ordered phase it is more ecient to up date the spins

with a heatbath algorithm for the measurements we decomp osed the spin

conguration into sto chastic SwendsenWang cluster and used the cluster

estimator h q i q hi j i where i j if i and j

s s

i j

b elong to the same cluster and otherwise All error bars are estimated

by means of the jackknife technique

Results

We rst checked that the average energy agrees with the exact result and

compared the second and third energy moments c V he

o

t o t

hei i and V he hei i with the large q expansions of Ref cp

o

Table Even for our smallest value q and the third moment we observe

a very go o d agreement with the Pade resummed large q expansion In addi

tion we lo oked at the magnetization m q hmaxfn giV q and its

i

cluster estimator m hjC j iV where n denotes the number of spins of

span i

orientation i q and jC j is the size of the largest spanning clus

span

ter Table shows that the two estimators give almost identical results which

Q

n n

x x agree very well with the exact expression m

n

where for q x is dened by q x x x Also

shown is the susceptibility m V hm hmi i and the third

o

o

moment m V hm hmi i

o

1

To conform with the present normalization our Monte Carlo estimate for

d

should b e multiplied by a factor of q

For the estimate of we concentrated on the k L pro jec

o t y y

tion g x The qualitative b ehaviour of g is illustrated in the semilog

plots of Fig For comparison also our previous results for g in

the disordered phase are shown Already these plots suggest that the two

correlation functions are governed by the same asymptotic decay law ie

that In fact the dotted lines interpolating the g data are

o t d t

constrained ts to the Ansatz assuming that

o d

for q To b e sure we also p er

formed unconstrained fourparameter ts using Ansatz Figure shows

that despite our high statistics in the ordered phase it was imp ossible to

get reliable estimates of g for very large distances Compared with our

analysis in the disordered phase the available t intervals are consequently

shifted to smaller x This makes the ts in the ordered phase somewhat more

sensitive to higherorder excitations and based on our previous exp erience

with varying t intervals we estimate that the numbers for collected in

o

Table should underestimate the true value by ab out For a com

parison of with we have therefore used approximately the same

o t d t

t intervals For example for q L L lattice and the t starting at

x we obtain from Table Recalling our estimate of

min o t

for a comparable t in the disordered phase this yields

d t

a ratio of suggesting again that This

o t d t o t d t

is corrob orated by the corresp onding estimates for q and q which

are actually closer to unity

To make this statement even more convincing we have plotted in Fig the

e e e

ratio where lng xg x is the usual eective correlation

o d

length with the correction already taken into account Here

o o

o

we have again implicitly assumed that higherorder excitations play a similar

role in b oth phases and eectively drop out when plotting the ratio of the two

correlation lengths Figure shows that over a sizeable range of distances

e e

with an uncertainty of only ab out We interpret this as

o d

further supp ort of our conjecture that

o t d t

Discussion

To summarize applying a nonzero momentum pro jection to the correlation

function in the ordered phase we determined with an accuracy compa

o t

rable to previous investigations in the disordered phase By comparing

the two correlation lengths at we obtain strong numerical evidence that

t

At rst sight this is in striking disagreement with a very recent

o d

exact pro of of the earlier conjecture for one denition of the or

o d

dered correlation length For another denition however only the

o o

relation could b e established in Ref This is clearly consistent

o o

with our result if we identify the numerically determined with We

o o

are currently investigating this problem in more detail by using precisely

the denitions of Ref which are based on geometrical prop erties of Potts

mo del clusters such as eg the distribution function of the cluster diameter

Acknowledgements

WJ thanks the DFG for a Heisenberg fellowship and SK gratefully acknowl

edges a fellowship by the Graduiertenkolleg Physik and Chemie supra

molekularer Systeme Work supp orted by computer grants hkf of HLRZ

J ulich and bvpf of Norddeutscher Vektorrechnerverbund NVV Berlin

HannoverKiel

References

For a recent overview see eg Dynamics of First Order Phase Transi

tions eds HJ Herrmann W Janke and F Karsch World Scientic

Singap ore

See eg JD Gunton MS Miguel and PS Sahni in Phase Transi

tions and Critical Phenomena Vol eds C Domb and JL Leb owitz

Academic Press New York K Binder Rep Prog Phys

T Bhattacharya R Lacaze and A Morel Nucl Phys B

T Bhattacharya R Lacaze and A Morel Nucl Phys B Pro c Suppl

T Bhattacharya R Lacaze and A Morel Europhys Lett

Nucl Phys B Pro c Suppl and private commu

nication

KC Lee Phys Rev Lett Seoul preprint cond

mat

W Janke in Computer Simulations in Condensed Matter Physics VII

eds DP Landau KK Mon and HB Schuttler Springer Verlag Hei

delb erg Berlin p

RB Potts Pro c Camb Phil So c

FY Wu Rev Mo d Phys E

FY Wu Rev Mo d Phys

E Buenoir and S Wallon J Phys A A Kl umper Int

J Mo d Phys B A Kl umper A Schadschneider and J

Zittartz Z Phys B

C Borgs and W Janke J Phys I France

BM Mc Coy and TT Wu The TwoDimensional Ising Model Harvard

University Press Cambridge Massachusetts C Itzykson and

JM Droue Statistical Field Theory Vol I Cambridge University

Press Cambridge

P Peczak and DP Landau Phys Rev B LA Fernan

dez JJ RuizLorenzo MP Lombardo and A Tarancon Phys Lett

B S Gupta and A Irback Phys Lett B

C Borgs and JT Chayes UCLA preprint September

W Janke and S Kappler Phys Lett A Nucl Phys B

Pro c Suppl

RJ Baxter J Phys A

W Janke and S Kappler to b e published

P (e) P (e)

o d

q = 10

L = 150

-1.6 -1.4 -1.2 -1

Figure Probability distribution P e in the two phases for L L dotted

line and L L lattices

Table Integrated auto correlation time of the energy and the number

inte

of up date sweeps in units of

inte

q q q

inte

L L

L L

Table Comparison of numerical and analytical results for energy moments

at in the ordered phase

t

Observable q q q

e MC L L

o

e MC L L

o

e exact

o

c MC L L

o

c MC L L

o

c large q

o

MC L L

o

MC L L

o

large q

o

Table Comparison of numerical and analytical results for magnetization

moments at in the ordered phase

t

Observable q q q

m MC L L

m MC L L

m MC L L

m MC L L

m exact

MC L L

o

MC L L

o

m MC L L

o

m MC L L

o

Table Numerical estimates of the correlation length from four

t

o

parameter ts to the Ansatz in the range x x For the L L

min max

lattices the ts along the x and y direction are distinguished by the index

q L q L q L

Lattice x x x x x x

max max max

o o o

min min min

L L

L L

x

L L

y

order MC Data L  L

order parat L  L

disorder MC Data L  L

disorder parat L  L

x

g

ln

q

L

order MC Data L  L

order parat L  L

disorder MC Data L  L

disorder parat L  L

x

g

ln

q

L

order MC Data L  L

order parat L  L

disorder MC Data L  L

disorder parat L  L

x

g

ln

q

L

x

Figure Semilogarithmic plots of the pro jected correlation function g

for q and in the ordered phase For comparison also g in the

disordered phase is shown

MC Data L  L

MC Data L  L

x

MC Data L  L

y e d

e o

q = 10

L = 150

MC Data L  L

MC Data L  L

x

MC Data L  L

y e d

e o

q = 15

L = 60

MC Data L  L

MC Data L  L

x

MC Data L  L

y e d

e o

q = 20

L = 40

x

Figure Ratio of eective correlation lengths in the ordered and disordered

phase for q and