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Physical Review A

Local-states method for the calculation of free energies in Monte Carlo simulations of lattice models

A. G. Schlijper, A. R. D. van Bergen, Berend Smit

Published Jan 1, 1990• 21 citations
Publisher

Abstract

We present and demonstrate an accurate, reliable, and computationally cheap method for the calculation of free energies in Monte Carlo simulations of lattice models. Even in the critical region it yields good results with comparatively short simulation runs. The method combines upper and lower bounds on the thermodynamic limit entropy density to yield not only an accurate estimate of the free energy but a bound on the possible error as well. The method is demonstrated on the two- and three-dimensional Ising models and the three-dimensional, three-states Potts model.

DOI
10.1103/physreva.41.1175

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We present and demonstrate an accurate, reliable, and computationally cheap method for the calculation of free energies in Monte Carlo simulations of lattice models. Even in the critical region it yields good results with comparatively short simulation runs. The method combines upper and lower bounds on the thermodynamic limit entropy density to yield not only an accurate estimate of the free energy but a bound on the possible error as well. The method is demonstrated on the two- and three-dimensional Ising models and the three-dimensional, three-states Potts model.
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We have performed Monte Carlo simulations for the three-dimensional Ising model. Using histogram techniques, we calculate the density of states on L3{L}^{3}L3 block lattices up to size L=14. Statistical jackknife methods are employed to perform a thorough error analysis. We obtain high-precision estimates for the leading zeros of the partition function, which, using finite-size scaling, translate into \ensuremath{\nu}=0.6285\ifmmode\pm\else\textpm\fi{}0.0019. Along a different line of approach following recent work in lattice-gauge theories, we accurately determine the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} correlation length) for cylindrical L2{L}^{2}L2Lz{L}_{z}Lz​ lattices (with Lz{L}_{z}Lz​=256 and L up to 12). The finite-size-scaling analysis of the mass-gap data leads to \ensuremath{\nu}=0.6321\ifmmode\pm\else\textpm\fi{}0.0019.
The study of the universal finite-size scaling amplitudes of interfacial free energies in Monte Carlo simulations provides an accurate method of determining the critical temperature and the universality class of phase transitions. These universal amplitudes can replace the critical exponents to identify universality classes. The authors use Bennett's optimisation scheme (1976) for the calculation of the interfacial free energies in Monte Carlo simulations. Tests for the Ising and three-state Potts model confirm the effectiveness of this method.
We present a simple method to estimate the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice model from Monte Carlo simulations. The main idea is to use the Wang-Landau flat-histogram algorithm, which allows us to obtain the free energy of a lattice model on a torus as a function of torus radii. The central charge is calculated with good precision from a free-energy scaling at the critical point. We apply the method to the Ising, tricritical Ising (Blume-Capel), Potts, and site-diluted Ising models, and we also discuss an estimation of the conformal weights.
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We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).