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Journal of Experimental and Theoretical Physics

Duality in two-dimensional Z(N)-symmetric spin models on a finite lattice

A. I. Bugriĭ, V. N. Shadura

Published Jan 1, 1998• 0 citations
Publisher

Abstract

We propose a method for deriving duality relations for two-dimensional inhomogeneous Z ( N )-symmetric models on a finite square lattice wound around a torus. The method is used to obtain duality relations for the vector Potts model, the Berezinskii-Villain Z ( N )-model, the Ashkin-Teller model, and the 8-vertex model on a lattice obliquely wound around a torus, as well as an exact relation linking the partition functions of the latter two models.

DOI
10.1134/1.558478

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We propose a method for deriving duality relations for two-dimensional inhomogeneous Z ( N )-symmetric models on a finite square lattice wound around a torus. The method is used to obtain duality relations for the vector Potts model, the Berezinskii-Villain Z ( N )-model, the Ashkin-Teller model, and the 8-vertex model on a lattice obliquely wound around a torus, as well as an exact relation linking the partition functions of the latter two models.
Duality relations for the two-dimensional nonhomogeneous Ising model on the finite square lattice wrapped on the torus are obtained. The partition function of the model on the dual lattice with arbitrary combinations of the periodical and antiperiodical boundary conditions along the cycles of the torus is expressed through some specific combination of the partition functions of the model on the original lattice with corresponding boundary conditions. It is shown that the structure of the duality relations is connected with the topological peculiarities of the dual transformation of the model on the torus.
Duality relations for the 2D nonhomogeneous Ising model on the finite square lattice wrapped on the torus are obtained. The partition function of the model on the dual lattice with arbitrary combinations of the periodical and antiperiodical boundary conditions along the cycles of the torus is expressed through some specific combination of the partition functions of the model on the original lattice with corresponding boundary conditions. It is shown that the structure of the duality relations is connected with the topological peculiarities of the dual transformation of the model on the torus.
An exact duality-decimation transformation is derived for a large class of two-dimensional spin systems on a square lattice. It maps a model with interactions in every second square (checkerboard interactions) onto a model with interactions in every square and vice versa. The transformation is applicable to Ising, Potts, solid-on-solid (SOS), and $x\ensuremath{-}y$ models. It may be used to reformulate the site percolation problem in terms of the $s\ensuremath{\rightarrow}1$ limit of a Potts model with four-spin interactions in every elementary square. The transformation maps the van Beijeren SOS model, which is equivalent to the FFF model, onto an $x\ensuremath{-}y$ model with checkerboard interactions. Thus one has an exactly soluble $x\ensuremath{-}y$ model, which, however, bears little resemblance to the ordinary $x\ensuremath{-}y$ model. The transformation may be combined with a bond-shifting approximation to obtain a real-space renormalization scheme which in some respects represents an improvement over the Migdal-Kadanoff formula. No nearest-neighbor bonds are shifted. Better values of the exponent $\ensuremath{\nu}$ are obtained for the Ising and Potts models. Apart from exponentially vanishing corrections, the renormalization transformation for SOS and $x\ensuremath{-}y$ models has a surface of fixed points corresponding to generalizations of the usual discrete Gaussian and Villain models.
The ferromagnetic q-states Potts model on a square lattice is analysed, for q > 4, through an elaborate version of the operatorial variational method. In the variational approach proposed in this paper, the duality relations are exactly satisfied, involving at a more fundamental level, a duality relationship between variational parameters. Besides some exact predictions, the approach is very effective in the numerical estimates over a wide range of temperature and can be systematically improved.
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.