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Journal of Physics C Solid State Physics

Decimation transformations for two-dimensional Ising spin systems

Leigh Sneddon, Michael N. Barber

Published Jul 28, 1977• 11 citations
Publisher

Abstract

The decimation transformation is used to calculate critical exponents for two-dimensional Ising spin systems on the square, triangular and hexagonal lattices. Both cumulant and cluster approximations are used. While the results of the low-order cumulant and of small cluster calculations agree well with the known exact exponents, an increase in cluster size does not appear to increase accuracy. The decimation transformation is only found to be useful for calculating thermal eigenvalues.

DOI
10.1088/0022-3719/10/14/017

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The decimation transformation is used to calculate critical exponents for two-dimensional Ising spin systems on the square, triangular and hexagonal lattices. Both cumulant and cluster approximations are used. While the results of the low-order cumulant and of small cluster calculations agree well with the known exact exponents, an increase in cluster size does not appear to increase accuracy. The decimation transformation is only found to be useful for calculating thermal eigenvalues.
The transformation is tested on the two-dimensional Ising model. The transformation is formed by compounding the decimation and Legendre transformations, as suggested by van Leeuwen (1975). The Legendre transformation involves an adjustable parameter p. The cluster approximation is used, as well as a decoupling approximation which is exact in the limit of p to 1. The procedure is seen to be capable of estimating both the thermal and magnetic critical exponents, v and eta respectively, for two-dimensional systems. The accuracy obtained is compared to that of other methods and the applicability of the procedure to three-dimensional systems is discussed.
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.
We have applied a decimation (dedecoration) method to a quantum system in order to obtain a critical exponent of the two-dimensional Ising model. Because the renormalization-group transformation is not exact, we have used a perturbation similar to the one used by Niemeijer and Van Leeuwen. The calculations were done to third order. Contrary to the 3-spin-cell method, the second order generates only next-nearest interaction, while the third order generates the next-next-nearest neighbor, etc. The numerical results are compared and analyzed. There are oscillations in our calculation, and the perturbation seems to be at least asymptotic.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.
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.