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arXiv:cond-mat/0509369 · Journal of Physics A Mathematical and General

Critical Binder cumulant in two-dimensional anisotropic Ising models

W Selke, Щур Лев Николаевич

Published Oct 19, 2005• 80 citations
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Abstract

The Binder cumulant at the phase transitions of Ising models on square lattices with various ferromagnetic nearest- and next-nearest-neighbour couplings is determined using mainly Monte Carlo techniques. We discuss the possibility of relating the value of the critical cumulant in the isotropic, nearest neighbour and in the anisotropic cases to each other by means of a scale transformation in rectangular geometry to pinpoint universal and nonuniversal features.

DOI
10.1088/0305-4470/38/44/l03

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The Binder cumulant at the phase transitions of Ising models on square lattices with various ferromagnetic nearest- and next-nearest-neighbour couplings is determined using mainly Monte Carlo techniques. We discuss the possibility of relating the value of the critical cumulant in the isotropic, nearest neighbour and in the anisotropic cases to each other by means of a scale transformation in rectangular geometry to pinpoint universal and nonuniversal features.
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.
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.
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.