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Physical Review E

Alternative boundary conditions for Monte Carlo simulations based on self-consistent correlations: Application to the two- and three-dimensional Ising models

I. Etxebarria, Cesar Capillas, Luis Elcoro

Published Mar 12, 2010• 0 citations
Publisher

Abstract

An alternative to periodic boundary conditions is developed and tested in Monte Carlo simulations of the two- and three-dimensional Ising models. The boundary conditions are based on a mean-field approach that incorporates consistency constraints for the magnetization and correlations between nearest neighbors by means of an effective field and an extra coupling between nearest neighbors at the boundary of the simulation box. During the simulation the self-consistent equations are solved, and statistics are accumulated to obtain thermodynamic averages. In comparison with the standard periodic boundary conditions the method gives a more accurate estimation of nonuniversal magnitudes, such as the transition temperature and the behavior of the magnetization, but it cannot compete with the accuracy of other strategies such as finite-size scaling theory or Monte Carlo renormalization group to obtain critical exponents.

DOI
10.1103/physreve.81.036703

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An alternative to periodic boundary conditions is developed and tested in Monte Carlo simulations of the two- and three-dimensional Ising models. The boundary conditions are based on a mean-field approach that incorporates consistency constraints for the magnetization and correlations between nearest neighbors by means of an effective field and an extra coupling between nearest neighbors at the boundary of the simulation box. During the simulation the self-consistent equations are solved, and statistics are accumulated to obtain thermodynamic averages. In comparison with the standard periodic boundary conditions the method gives a more accurate estimation of nonuniversal magnitudes, such as the transition temperature and the behavior of the magnetization, but it cannot compete with the accuracy of other strategies such as finite-size scaling theory or Monte Carlo renormalization group to obtain critical exponents.
Results are presented for the computer simulation of the time-dependent behavior of the Ising model on a 100 \ifmmode\times\else\texttimes\fi{} 100 lattice. Groups of particles are regarded as samples from an infinite lattice and the interactions between the sample particles and those outside are expressed through new boundary conditions based on extending the concept of mean field in a self-consistent way. These new boundary conditions are explored analytically for a few small samples and a sharp critical point is found in contrast to periodic boundary conditions. The relaxation times for long-range order and nearest-neighbor correlation (energy) for the 100 \ifmmode\times\else\texttimes\fi{} 100 system have been computed and the corresponding critical exponents estimated. The results are ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}=2.30\ifmmode\pm\else\textpm\fi{}0.30$ and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}}=0.38\ifmmode\pm\else\textpm\fi{}0.04$; these are compared with other recent results from computer simulations and series calculations.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
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.
We describe a hierarchy of stochastic boundary conditions (SBCs) that can be used to systematically eliminate finite size effects in Monte Carlo simulations of Ising lattices. For an Ising model on a 100×100100 \times 100100×100 square lattice, we measured the specific heat, the magnetic susceptibility, and the spin-spin correlation using SBCs of the two lowest orders, to show that they compare favourably against periodic boundary conditions (PBC) simulations and analytical results. To demonstrate how versatile the SBCs are, we then simulated an Ising lattice with a magnetized boundary, and another with an open boundary, measuring the magnetization, magnetic susceptibility, and longitudinal and transverse spin-spin correlations as a function of distance from the boundary.
Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling JNN{J}_{\mathrm{NN}}JNN​ between nearest neighbors and additional interactions JNNN{J}_{\mathrm{NNN}}JNNN​ between next-nearest neighbors and J3NN{J}_{3\mathrm{N}\mathrm{N}}J3NN​ between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=JNNN/JNN{J}_{\mathrm{NNN}/{J}_{\mathrm{NN}}}JNNN/JNN​​ and R'=J3NN/JNN{J}_{3\mathrm{N}\mathrm{N}/{J}_{\mathrm{NN}}}J3NN/JNN​​. In addition to the c(2\ifmmode\times\else\texttimes\fi{}2) and (2\ifmmode\times\else\texttimes\fi{}1) phases, which also occur for R'=0, we find new (4\ifmmode\times\else\texttimes\fi{}4) and (4\ifmmode\times\else\texttimes\fi{}2) ordered states, for R'\ensuremath{\ne}0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2\ifmmode\times\else\texttimes\fi{}1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.
A generalization of periodic boundary conditions to incorporate fluctuations with wavelength larger than the system size is discussed. These boundary conditions are applied to Monte Carlo calculations on the two-dimensional XY\mathrm{XY}XY model. The finite-size effect on the vortex-vortex correlation function, and thereby the dielectric function, is shown to be virtually eliminated. These Monte Carlo data are shown to be in excellent agreement with the Kosterlitz renormalization group equations, and the critical temperature is found to be Tc=0.8922(2){T}_{c}=0.8922(2)Tc​=0.8922(2).