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Journal of Physics A Mathematical and General

Self-duality relation for a three-dimensional spin-1/2lattice system

Héctor Giacomini

Published Jun 21, 1985• 1 citation
Publisher

Abstract

A three-dimensional system consisting of a set of interacting two-dimensional Ising models is introduced. It is shown that this model is self-dual. It contains the three-dimensional Ising model as a particular case.
DOI
10.1088/0305-4470/18/9/037

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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 We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m . Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H . The 2D system, with size m × N / m , has the topology of a cylinder with helical BC. In the thermodynamic limit N / m → ∞ , m → ∞ , a 2D critical singularity develops on the self-duality line, sinh 2 K sinh 2 H = 1 .
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.
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.