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Journal of Physics A Mathematical and General

Cluster expansions for the Ising and Heisenberg spin models in Hamiltonian lattice field theory

Chris J. Hamer, Anthony John Guttmann

Published Sep 7, 1989• 9 citations
Publisher

Abstract

Cluster expansion methods are applied to calculate 'high-temperature' series for the vacuum energy, the susceptibility and the mass gap for the Ising model and the O(2) and O(3) Heisenberg models in (1+1) dimensions and (2+1) dimensions. Critical points and critical indices are estimated for the line, the square and the triangular lattices. The results demonstrate universality with the normal Euclidean versions of these models, within errors.

DOI
10.1088/0305-4470/22/17/030

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Annual Conference on Magnetism and Magnetic Materials (MMM)•Jan 1, 1974•0 citations
Cluster expansion methods are applied to calculate 'high-temperature' series for the vacuum energy, the susceptibility and the mass gap for the Ising model and the O(2) and O(3) Heisenberg models in (1+1) dimensions and (2+1) dimensions. Critical points and critical indices are estimated for the line, the square and the triangular lattices. The results demonstrate universality with the normal Euclidean versions of these models, within errors.
High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.
We have derived series in powers of K to 8‐th order for the two‐point function, susceptibility and free energy of the model Hamiltonian, ΣR⇒ {W[Q(R⇒)] + K/2 Σδ⇒ Q(R⇒)·Q(R⇒+δ⇒)}, where Q is a real tensor, and W is an arbitrary, even function of Q. Special cases include the spin‐S Ising model and the classical Heisenberg model, for both of which W = 0. Additional applications include models for structural phase transitions, tricritical points, He3‐He4 mixtures, liquid crystals, and the class of models employed in renormalization — group studies of critical phenomena. Detailed evaluation of susceptibility series have been carried out on the standard 2, 3, and 4‐dimensional lattices. The series for the scalar case on the triangular net is presented herein.