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Progress of Theoretical Physics

Renormalization-Group Transformation of Three-Dimensional Ising Models

T. Tatsumi

Published Mar 1, 1978• 6 citations
Publisher

Abstract

Ising models on the square and simple cubic lattices are studied by a slightly modified version of the Niemeijer-van Leeuwen renormalization-group transformation. For the simple cubic lattice, we find the transition temperature Kc = 0.226 with critical exponents α= -0.420 and δ= 5.04 from the third order cumulant expansion. Ising models are also treated by the two-, three- and four-step decimation transformations among the simple cubic, body-centered cubic and face-centered cubic lattices. It is found from the three-step transformations that three systems have the same critical exponents. In contrast to the results from the cumulant expansion method, this method yields the positive specific heat exponent except one case. These results are compared with the known series values.

DOI
10.1143/ptp.59.714

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Ising models on the square and simple cubic lattices are studied by a slightly modified version of the Niemeijer-van Leeuwen renormalization-group transformation. For the simple cubic lattice, we find the transition temperature Kc = 0.226 with critical exponents α= -0.420 and δ= 5.04 from the third order cumulant expansion. Ising models are also treated by the two-, three- and four-step decimation transformations among the simple cubic, body-centered cubic and face-centered cubic lattices. It is found from the three-step transformations that three systems have the same critical exponents. In contrast to the results from the cumulant expansion method, this method yields the positive specific heat exponent except one case. These results are compared with the known series values.
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.
We demonstrate the applicability of the εεε-convergence algorithm in extracting the critical temperatures and critical exponents of three-dimensional Ising models. We analyze the low temperature magnetization as well as high temperature susceptibility series of simple cubic, body-centered cubic, face-centered cubic and diamond lattices, using two different variables for the inverse critical temperature. In the case of simple cubic lattices, the magnetization series was modified to deduce accurate values of the critical temperatures. The alternate variable for dimensionless inverse temperature suggested by Guttmann and Thompson has also been employed for the estimation of the critical parameters.
A real space renormalisation group method has been applied to find the critical temperature of the Ising model on cubic and octahedral lattices. The result obtained for a cubic lattice is in good agreement with that obtained from high-temperature series expansion and is better than the results of other papers. A modification of the transformation tested for a simple cubic lattice allowed the authors to find the critical temperature for an octahedral lattice. The latter has not so far been studied by any version of the real space renormalisation group method.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.