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Journal of Physics A Mathematical and General

The Kadanoff lower-bound variational renormalization group applied to an SU(2) lattice spin model

Guðmar Þorleifsson, P.H. Damgaard

Published Dec 21, 1990• 3 citations
Publisher

Abstract

The authors apply the variational lower-bound renormalization group transformation of Kadanoff (1975) to an SU (2) lattice spin model in two and three dimensions. Even in the one-hypercube framework of this renormalization group transformation the present model is characterized by having an infinite basis of fundamental operators. They investigate whether the lower-bound variational renormalization group transformation yields results stable under different truncations of this operator basis. Their results show that for this particular spin model this is not the case.

DOI
10.1088/0305-4470/23/24/026

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The authors apply the variational lower-bound renormalization group transformation of Kadanoff (1975) to an SU (2) lattice spin model in two and three dimensions. Even in the one-hypercube framework of this renormalization group transformation the present model is characterized by having an infinite basis of fundamental operators. They investigate whether the lower-bound variational renormalization group transformation yields results stable under different truncations of this operator basis. Their results show that for this particular spin model this is not the case.
The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic ddd-dimensional lattice for all integral values of z=2dz={2}^{d}z=2d between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.
The fixed-point topology of Kadanoff's lower-bound renormalization transformation is discussed for the d=2d=2d=2 square-lattice Ising model in the three-dimensional space of coupling constants for the nearest-neighbor, next-nearest neighbor, and four-spin interactions. An obstacle one encounters in trying to calculate a critical surface is pointed out. In the three-dimensional space of coupling constants Kadanoff's fixed point has two relevant vectors and can only be reached from a critical line, not from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. The critical exponents of this fixed point are close to the exact values, though not nearly so close as for Kadanoff's fixed point. The properties of both fixed points as a function of the variational parameter ppp are described. Critical values of the nearest-neighbor couplings leading to the fixed points are compared.
A Kadanoff lower-bound renormalization transformation with three variational parameters has been applied to the Blume-Capel model by Burkhardt. We report some additional results obtained in a particular one-parameter subspace of the three variational parameters. In this subspace the transformation treats the three possible Blume-Capel ground-state configurations with all the spins in the same state on an equal footing and preserves the three-state permutational symmetry of the Potts model. There are fixed points associated with first-order, critical, and tricritical transitions. The variational parameters maximizing the free energy at the critical and tricritical fixed points differ by only a few percent. The critical exponents associated with the different transitions and various points on the critical surfaces are calculated for d=2,3d=2,3d=2,3 dimensions.
From a consideration of the ground states of the antiferromagnetic Ising model a weighting function is proposed. This function preserves the ground states under the renormalisation group transformation. This form of the lower bound approximation leads to results which are in excellent agreement with the exact values in two dimensions and in fair agreement with the approximate results of the series expansion in three dimensions. The implication of this approach for other real-space approximations is noted.
The use of variational approximations in the study and application of renormalization groups is discussed. In particular, a simple approximation is derived which yields an upper bound to the free energy of Ising models on d-dimensional lattices. The optimal transformation, which yields the least upper bound, is determined analytically. The criterion proposed by Kadanoff (1975) to determine the 'best' approximation to the fixed point is found to fail in this case. The reasons for this failure and several of the basic problems posed by variational approximations are discussed.