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Journal of Physics A Mathematical and General

Duality transformations for spin-1/2lattice systems

Héctor Giacomini

Published Jun 21, 1985• 2 citations
Publisher

Abstract

By using a procedure introduced in the previous work, the author considers duality transformations for spin-1/2 lattice systems. The author shows that, in some cases, given a self-dual model, a new self-dual model can be obtained from it. Several examples of this fact are given. In particular the author analyses a system consisting of two interacting Baxter-Wu models, which present close analogies with the Ashkin-Teller model.

DOI
10.1088/0305-4470/18/9/030

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By using a procedure introduced in the previous work, the author considers duality transformations for spin-1/2 lattice systems. The author shows that, in some cases, given a self-dual model, a new self-dual model can be obtained from it. Several examples of this fact are given. In particular the author analyses a system consisting of two interacting Baxter-Wu models, which present close analogies with the Ashkin-Teller model.
The zero-temperature renormalization-group transformations for quantum spin systems are analyzed. The block transformation and the decimation-type transformation used in the study of the one-dimensional Ising model in transverse field are extended to the quantum version of the Potts model and Ashkin-Teller model. It is shown that for these self-dual models the two kinds of renormalization-group transformations are dual to each other and therefore give the same result for the critical behavior. The duality persists even if the higher-lying states are taken into account in a perturbational way.
Properties of general lattice systems are investigated using the group properties of such systems. The duality transformation is generalized and is expressed in terms of exact short sequence between the groups associated with one lattice and the groups associated with the dual lattice. A general method is given for constructing explicitly a dual lattice for any lattice system and is illustrated with some two- and three-dimensional models. In particular it is shown that the two-dimensional triangular lattice with three-body forces is self-dual.
Considers the pure Z(N) spin systems (including the standard Ising and Potts models) as well as generalised gauge systems (plaquettes or more complex simplex) in d-dimensional hypercubic lattices. These models are self-dual, and the authors show how this duality can be thought of as a series-parallel transformation. The simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems, including some particular asymptotic behaviours which are exact. As an illustration the d=2 diluted Z(4) spin system is discussed in some detail: for those regions where exact results are available the agreement is satisfactory.