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Physical review. B, Solid state

Decimation method for two-dimensional Ising exponents

Gregory S. Um

Published Mar 1, 1977• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.15.2736

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Physical review. B, Solid state•Feb 1, 1976•23 citations
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.
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.
The Niemeijer-van Leeuwen formulation of the renormalisation group has been used to discuss the quenched dilute Ising model and the Ising spin-glass in two dimensions. The authors chose a five-spin cell which, in the pure case, treats the Ising antiferromagnet equally as well as the ferromagnet. This choice of cell leads to the existence of a spin-glass ground state which was found to be stable for the whole composition range. The spin-glass transition was found to be sharp and the corresponding critical exponents were calculated. For the dilute Ising model the behaviour of the reduced critical temperature T c (p)/T c (1), the percolation concentration and the correlation length exponent were found to be in good agreement with the exact results and series calculations.
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.