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Physical review. B, Condensed matter

Monte Carlo studies of critical free energies and the simple-cubic Ising model

K. K. Mon

Published Jan 1, 1989• 25 citations
Publisher

Abstract

This paper addresses the limitation for routine applications of the proposed Monte Carlo method of direct calculation of absolute free energy by sampling of finite-size dependences and discusses new extensions. The need for multistage sampling is removed and the method is now simpler to apply in practice. To exhibit this, the critical free-energy density for the simple-cubic nearest-neighbor ferromagnetic Ising model is studied for a wide range of lattice sizes, up to 32\ifmmode\times\else\texttimes\fi{}32\ifmmode\times\else\texttimes\fi{}32. The finite-size scaling is demonstrated and the scaling amplitude calculated with higher accuracy than before. These results indicate the possibility that very accurate free-energy density and related quantities such as surface free energy, can be routinely obtained for lattice models from Monte Carlo simulations, even at criticality.

DOI
10.1103/physrevb.39.467

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It is shown that the absolute free energy of lattice spin systems can be calculated directly by a novel application of Monte Carlo sampling of finite-size dependence. Results are obtained for the two- and three-dimensional Ising models at Tc{T}_{c}Tc​ and are consistent with recent finite-size scaling-theory predictions. The free energy is in excellent agreement with Onsager's exact solution for two dimensions and with series expansions for three dimensions. The U0{U}_{0}U0​ is measured for the first time. Results are consistent with the proposed universality.
We present an extensive Monte Carlo study of the three-dimensional Ising model with free surfaces, edges, and corners at bulk criticality. Finite-size scaling properties of the excess free energies of free surfaces, edges, and corners are estimated by use of the recently proposed applications of the parametric integration method. The corner excess free energies are obtained for the first time and are consistent with the logarithmic finite-size scaling predictions. Results of the simplecubic and body-centered-cubic lattices provide support for the proposed universality of the finite-size scaling amplitudes for the excess free energies of free surfaces, edges, and corners.
A transfer-matrix Monte Carlo technique is developed to compute the free energy of three-dimensional, classical Heisenberg ferromagnets. From the free energy of systems with periodic and antiperiodic boundary conditions, helicity moduli are calculated. From these the critical couplings for simple-cubic (sc) and face-centered-cubic lattices are estimated by use of finite-size scaling. For the simple-cubic lattice, the critical dimension of the surface magnetization is estimated with standard Monte Carlo methods, yielding a result in excellent agreement with the \ensuremath{\epsilon}-expansion work of Diehl and N\"usser.
In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.