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Journal of Physics A Mathematical and General

A variational real-space renormalisation-group transformation based on the cumulant expansion

Scott J. Shenker, Leo P. Kadanoff, Adrianus M. M. Pruisken

Published Jan 1, 1979• 12 citations
Publisher

Abstract

The first order cumulant expansion of Niemeijer and van Leeuwen (1974) is combined with a variational method to find the 'best' value of a fixed point Hamiltonian depending upon a parameter, p. A new method for calculating eigen values near this fixed point is proposed, based upon the neglect of terms proportional to the error neglected at the fixed point. This approach determines delta p/ delta K unambiguously. Calculations are carried out for a few simple models. The results are more accurate than those of most other comparably simple approaches.

DOI
10.1088/0305-4470/12/1/021

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Communications in Theoretical Physics•Jan 30, 1999•3 citations
The first order cumulant expansion of Niemeijer and van Leeuwen (1974) is combined with a variational method to find the 'best' value of a fixed point Hamiltonian depending upon a parameter, p. A new method for calculating eigen values near this fixed point is proposed, based upon the neglect of terms proportional to the error neglected at the fixed point. This approach determines delta p/ delta K unambiguously. Calculations are carried out for a few simple models. The results are more accurate than those of most other comparably simple approaches.
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.
The two-dimensional XY\mathrm{XY}XY and Heisenberg models are studied with a quantum-mechanical generalization of the Niemeijer and Van Leeuween renormalization-group method. In second-order cumulant expansion a fixed point exists for the XY\mathrm{XY}XY model, giving a critical temperature at $\frac{J}{k{T}_{c}}\ensuremath{\cong}0.87\ifmmode\pm\else\textpm\fi{}0.05$. For the Heisenberg model we find a fixed point at K∗=2.8{K}^{*}=2.8K∗=2.8. This result may not be reliable, since the truncation of the cumulant expansion is only valid for small KKK.
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.