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2D Potts Model Correlation Lengths KOMA Revised version May D Potts Mo del Correlation Lengths Numerical Evidence for = at o d t Wolfhard Janke and Stefan Kappler Institut f ur Physik Johannes GutenbergUniversitat Mainz Staudinger Weg Mainz Germany Abstract We have studied spinspin correlation functions in the ordered phase of the twodimensional q state Potts mo del with q and at the rstorder transition p oint Through extensive Monte t Carlo simulations we obtain strong numerical evidence that the cor relation length in the ordered phase agrees with the exactly known and recently numerically conrmed correlation length in the disordered phase As a byproduct we nd the energy moments o t t d in the ordered phase at in very go o d agreement with a recent large t q expansion PACS numbers q Hk Cn Ha hep-lat/9509056 16 Sep 1995 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
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