arXiv:hep-th/9706045

Duality in 2D Spin Models on Torus

A. I. Bugrij, V. N. Shadura

Published Jun 6, 1997• 1 citation

Abstract

Method of derivation of the duality relations for two-dimensional Z(N)-symmetric spin models on finite square lattice wrapped on the torus is proposed. As example, exact duality relations for the nonhomogeneous Ising model (N=2) and the Z(N)-Berezinsky-Villain model are obtained.

DOI: 10.48550/arxiv.hep-th/9706045

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arXiv:hep-th/9706045

Duality in 2D Spin Models on Torus

A. I. Bugrij, V. N. Shadura

Published Jun 6, 1997• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.hep-th/9706045

Related papers

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Journal of Experimental and Theoretical PhysicsJan 1, 19980 citations

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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.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.

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The Kramers-Wannier duality is shown to hold for all the even number spin correlation functions of the two dimensional square lattice Ising model in the sense that the high temperature

(T>T_{c})

expressions for these correlation functions are transformed into the low temperature $(T

F

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.