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Physical review. B, Solid state

Study of the cumulant expansion for renormalization transformations on two- and three- dimensional Ising models

Shin-chieh Hsu, J. D. Gunton

Published Mar 1, 1977• 25 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.15.2688

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Physical Review Letters•May 31, 1976•51 citations
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.
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 (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.
We have calculated the three-spin cell cumulant expansion of Niemeijer and Van Leeuwen for the one-dimensional XY\mathrm{XY}XY model in order to obtain the critical exponents of the two-dimensional Ising model. We found that the calculation of this type is not as accurate as the direct cumulant-expansion calculation for the two-dimensional Ising model. Specifically, the results of our present calculation deviate considerably from the exact results in comparison to the results obtained by Hsu, Niemeijer, and Gunton. Moreover, we find that the odd-spin eigenvalue is not physical in this particular renormalization-group transformation.
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.