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Physical Review Letters

Critical Surface for Square Ising Spin Lattice

Michael Nauenberg, Bernard Nienhuis

Published Oct 14, 1974• 140 citations
Publisher

Abstract

The renormalization-group transformations for a square Ising spin lattice are evaluated in a four-cell-cluster approximation to obtain the critical surface in a three-dimensional space of spin-interaction parameters. Agreement is found with known critical curves, particularly Baxter's analytic solution.
DOI
10.1103/physrevlett.33.944

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We suggest that the Monte Carlo renormalization group, when combined with the type of cell-spin transformation introduced by van Leeuwen, should be a powerful tool in the study of Ising models with n> 2n>~2n> 2. Numerical results are presented for the Baxter model and the Ising model with nearest- and next-nearest-neighbor interactions on a square lattice.
In evaluating the critical properties of lattice spin systems in the real-space renormalization-group theory we use the cluster variation method. A configuration in the transformed system is constrained and the probability of occurrence of this configuration is calculated both in the transformed system and in the original system. By equating the two probabilities and forming ratios of two such equalities (for two or more constrained configurations) the fixed point of the renormalization transformation is evaluated. The method can avoid the trouble due to different singularities in the original and transformed systems, and hence can obviate the possible development of spurious singularities in the transformation at low temperatures. The two-dimensional triangular Ising model is treated with numerical results comparable with those obtained by the cluster treatment of Niemeijer and van Leeuwen who used more and larger cluster types than those we introduce.