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arXiv:cond-mat/0701515 · Journal of Statistical Mechanics Theory and Experiment

The critical Binder cumulant for isotropic Ising models on square and triangular lattices

Walter Selke

Published Apr 18, 2007• 44 citations
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Abstract

Using Monte Carlo techniques, the critical Binder cumulant U * of isotropic nearest neighbour Ising models on square and triangular lattices is studied. For rectangular shapes, employing periodic boundary conditions, U * is found to show the same dependence on the aspect ratio for both lattice types. Similarly, applying free boundary conditions for systems with square as well as circular shapes for both lattices, the simulational findings are also consistent with the suggestion that, for isotropic Ising models with short range interactions, U * depends on the shape and the boundary condition, but not on the lattice structure.

DOI
10.1088/1742-5468/2007/04/p04008

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Using Monte Carlo techniques, the critical Binder cumulant U * of isotropic nearest neighbour Ising models on square and triangular lattices is studied. For rectangular shapes, employing periodic boundary conditions, U * is found to show the same dependence on the aspect ratio for both lattice types. Similarly, applying free boundary conditions for systems with square as well as circular shapes for both lattices, the simulational findings are also consistent with the suggestion that, for isotropic Ising models with short range interactions, U * depends on the shape and the boundary condition, but not on the lattice structure.
We reanalyze transfer-matrix and Monte Carlo results for the critical Binder cumulant U* of an anisotropic two-dimensional Ising model on a square lattice in a square geometry with periodic boundary conditions. Spins are coupled between nearest-neighboring sites and between next-nearest-neighboring sites along one of the lattice diagonals. We find that U* depends only on the asymptotic critical long-distance features of the anisotropy, irrespective of its realization through ferromagnetic or antiferromagnetic next-nearest-neighbor couplings. We modify an earlier renormalization-group calculation to obtain a quantitative description of the anisotropy dependence of U*. Our results support our recent claim towards the validity of universal finite-size scaling for critical phenomena in the presence of a weak anisotropy.
We investigate the dependence of the critical Binder cumulant of the magnetization and the largest Fortuin-Kasteleyn cluster on the boundary conditions and aspect ratio of the underlying square Ising lattices. By means of the Swendsen-Wang algorithm, we generate numerical data for large system sizes and we perform a detailed finite-size scaling analysis for several values of the aspect ratio r, for both periodic and free boundary conditions. We estimate the universal probability density functions of the largest Fortuin-Kasteleyn cluster and we compare it to those of the magnetization at criticality. It is shown that these probability density functions follow similar scaling laws, and it is found that the values of the critical Binder cumulant of the largest Fortuin-Kasteleyn cluster are upper bounds to the values of the respective order-parameter's cumulant, with a splitting behavior for large values of the aspect ratio. We also investigate the dependence of the amplitudes of the magnetization and the largest Fortuin-Kasteleyn cluster on the aspect ratio and boundary conditions. We find that the associated exponents, describing the aspect-ratio dependencies, are different for the magnetization and the largest Fortuin-Kasteleyn cluster, but in each case are independent of boundary conditions.
The Binder cumulant at the phase transition of Ising models on square lattices with ferromagnetic couplings between nearest neighbors and with competing antiferromagnetic couplings between next-nearest neighbors, along only one diagonal, is determined using Monte Carlo techniques. In the phase diagram a disorder line occurs separating regions with monotonically decaying and with oscillatory spin-spin correlations. Findings on the variation of the critical cumulant with the ratio of the two interaction strengths are compared to related recent results based on renormalization-group calculations.