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Physical Review Letters

Monte Carlo Renormalization Group and Ising Models with n>~2

Robert H. Swendsen, S. Krinsky

Published Jul 16, 1979• 145 citations
Publisher

Abstract

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 $n>~2$. Numerical results are presented for the Baxter model and the Ising model with nearest- and next-nearest-neighbor interactions on a square lattice.

DOI
10.1103/physrevlett.43.177

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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.
The effectiveness of a Monte Carlo renormalization-group method is studied by applying it to the Baxter-Wu model (Ising spins on a triangular lattice with three-spin interactions). The calculations yield three relevant eigenvalues in good agreement with exact or conjectured results. We demonstrate that the method is capable of distinguishing between models expected to be in the same universality class, when one of them (four-state Potts) exhibits logarithmic corrections to the usual power-law singularities and the other (Baxter-Wu) does not.
The Niemeijer-van Leeuwen (1973) cumulant expansion for calculation of renormalized interactions is carried to third order for the Ising model on the triangular lattice. Kadanoff cells containing three spins are used. Five types of pair interactions and three types of four-spin interaction enter the calculations. The renormalization equations have one non-trivial fixed point. The accuracy of the result for the corresponding relevant eigenvalues is, however, less than in second order. The idea that the cumulant expansion is an asymptotic expansion has thus been given numerical support.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
The Niemeijer-Van Leeuwen cumulant expansion is used in the renormalization-group analysis of the critical-point behavior of the Ising model. Second- and third-order results are obtained for the simple cubic and square lattices, respectively, for a variety of cell sizes, using a parameter-dependent renormalization transformation introduced by Kadanoff and Houghton. Although the results in two dimensions are in general good, there is no evidence of any convergence, with the best values for the critical exponents and critical temperature being given by the second-order theory. In three dimensions the values found for the critical temperature and magnetic scaling index for a cell of 27 spins are reasonable, but the corresponding thermal scaling index is quite poor.