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International Journal of Modern Physics C

MULTICANONICAL SIMULATION OF THE TWO-DIMENSIONAL 7-STATE POTTS MODEL

Wolfhard Janke E

Published Oct 1, 1992• 60 citations
Publisher

Abstract

Using multicanonical Monte Carlo simulations, we have calculated the interface free energy between the coexisting ordered and disordered phase at the temperature driven first-order phase transition of the two-dimensional 7-state Potts model. Our result 2f3= 0.0241±0.0010 is about an order of magnitude smaller than other estimates in the recent literature. We discuss the differences in analysis and give a possible explanation for the discrepancy. Finally, we briefly mention similar investigations at the deconfining phase transition of SU(3) lattice gauge theory.

DOI
10.1142/s0129183192000762

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The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.
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Relying on the recent proposed multicanonical algorithm, we present a numerical simulation of the first-order phase transition in the 2D 10-state Potts model on lattices up to sizes 100\ifmmode\times\else\texttimes\fi{}100. It is demonstrated that the new algorithm lacks an exponentially fast increase of the tunneling time between metastable states as a function of the linear size L of the system. Instead, the tunneling time diverges approximately proportional to L2.65{\mathit{L}}^{2.65}L2.65. On our largest lattice we gain more than 2 orders of magnitude as compared to a standard heat-bath algorithm. As a first physical application we report a high-precision computation of the interfacial free energy per unit area.
We have calculated the large-qqq series of the energy cumulants, the magnetization cumulants and the correlation length at the first order phase transition point both in the ordered and disordered phases for the qqq-state Potts model in two dimensions. The series enables us to estimate the numerical values of the quantities more precisely by a factor of 102−10410^2 - 10^4102−104 than the Monte Carlo simulations. From the large-qqq series of the eigenvalues of the transfer matrix, we also find that the excited states form a continuum spectrum and there is no particle state at the first order phase transition point.
We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point βtβ_tβt​. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.