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Physical review. B, Condensed matter

Simulation of an ensemble with varying magnetic field: A numerical determination of the order-order interface tension in theD=2 Ising model

Bernd A. Berg, Ulrich H. E. Hansmann, T. Neuhaus

Published Jan 1, 1993• 107 citations
Publisher

Abstract

In analogy with a recently proposed multicanonical ensemble we introduce an ensemble where the partition function is simulated with a term in the action containing a varying magnetic field. Using this ensemble we demonstrate on lattices with periodic boundary conditions that it is possible to enhance the appearance of order-order interfaces by many orders of magnitude. To perform a stringent test of the method we consider the D=2 Ising model at \ensuremath{\beta}=0.5 and simulate square lattices up to size 100\ifmmode\times\else\texttimes\fi{}100. By a finite-size scaling analysis, the order-order interface tension per unit area is obtained. Our best infinite-volume extrapolation is in excellent agreement with Onsager's exact result.

DOI
10.1103/physrevb.47.497

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In analogy with a recently proposed multicanonical ensemble we introduce an ensemble where the partition function is simulated with a term in the action containing a varying magnetic field. Using this ensemble we demonstrate on lattices with periodic boundary conditions that it is possible to enhance the appearance of order-order interfaces by many orders of magnitude. To perform a stringent test of the method we consider the D=2 Ising model at \ensuremath{\beta}=0.5 and simulate square lattices up to size 100\ifmmode\times\else\texttimes\fi{}100. By a finite-size scaling analysis, the order-order interface tension per unit area is obtained. Our best infinite-volume extrapolation is in excellent agreement with Onsager's exact result.
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.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.
In a recent paper, Clusel and Fortin ( 2006 J. Phys. A: Math. Gen. 39 995 ) presented an analytical study of a first-order transition induced by an inhomogeneous boundary magnetic field in the two-dimensional Ising model. They identified the transition that separates the regime where the interface is localized near the boundary from that where it propagates inside the bulk. Inspired by these results, we measured the interface tension by using multimagnetic simulations combined with parallel tempering to determine the phase transition and the location of the interface. Our results are in very good agreement with the theoretical predictions. Furthermore, we studied the spin–spin correlation function for which no analytical results are available.
We use the transfer-matrix method to calculate density-of-states (DOS) functions for fully finite Ising-model systems. In this way, we obtain expressions for the corresponding partition functions for systems of sizes up to 12\ifmmode\times\else\texttimes\fi{}12 in zero field and up to 8\ifmmode\times\else\texttimes\fi{}8 in nonzero external magnetic field. We also establish a way of representing the DOS functions, in terms of the suitably scaled variables, that reveals a remarkably rapid convergence of these functions towards their thermodynamic limiting values as the system size increases. This rapid convergence implies that results obtained for systems of small sizes contain a surprisingly large amount of information about the behavior of infinite systems. In particular, we demonstrate that the finite-size-scaling-theory predictions are satisfied for small systems (treated exactly via this method) comparably well as for much larger systems (treated approximately via the Monte Carlo method). In addition, applying the method to the square Ising model in a nonzero field, we discover that the DOS functions for small systems display a regularity that one would expect to appear only for very large systems.