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Progress of Theoretical Physics

A Duality between Blocked Ising Models

Y. Kasai

Published Oct 1, 1978• 0 citations
Publisher

Abstract

For planar lattices, there exists a dual relation between extended (blocked) Ising models in which interactions are expressed by face elements (blocks) instead of bonds. The total interaction for a block contains all even spin products among spins on the face (g-polygon); for g = 2 or 3, it contains only two-spin product, for g = 4 it contains crossed bond interactions and four-spin product, and so on. Formally, there exists a self-dual lattice for every g. The triangular lattice is self-dual for g = 3 and the self-dual lattice for g ≥4 has non integer coordinate number (a rational number) which suggests a universality for the coordination number.

DOI
10.1143/ptp.60.958

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For planar lattices, there exists a dual relation between extended (blocked) Ising models in which interactions are expressed by face elements (blocks) instead of bonds. The total interaction for a block contains all even spin products among spins on the face (g-polygon); for g = 2 or 3, it contains only two-spin product, for g = 4 it contains crossed bond interactions and four-spin product, and so on. Formally, there exists a self-dual lattice for every g. The triangular lattice is self-dual for g = 3 and the self-dual lattice for g ≥4 has non integer coordinate number (a rational number) which suggests a universality for the coordination number.
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.
Abstract These lecture notes deal with Ising models with interactions containing products of more than two spins. After a number of examples have been given, some general statements are presented. Of particular interest is a gauge-invariant Ising model in four dimensions. This has important properties in common with models for quantum chromodynamics as developed by Ken Wilson. One phase yields an area law for the Wilson loop, yielding an interaction that increases proportionally to the distance and thus corresponds to quark confinement. The other phase yields a perimeter law allowing for a quark–gluon plasma.
It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising models Mdn on d-dimensional lattices characterized by a number n = 1, 2, … , d (n = 1 corresponds to the Ising model with two-spin interaction). These models are related by two duality transformations. The models with 1 < n < d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate the existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long-range order of the spins in the low temperature phase.