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Journal of Physics A Mathematical and General

The exact solution of a nonplanar Ising model

Roger Bowers

Published Oct 1, 1983• 0 citations
Publisher

Abstract

A generalisation of the star-triangle transformation is used to establish a correspondence between a nonplanar Ising lattice with two interaction constants and the standard triangular Ising model. The critical point of the nonplanar lattice is obtained exactly for various values of the ratio of the interaction constants. The singularity in the zero-field free energy of the nonplanar lattice is investigated. It is shown to be of the same form as that displayed by the corresponding standard Ising model.

DOI
10.1088/0305-4470/16/14/008

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Modern Physics Letters B•Feb 28, 1996•2 citations
A generalisation of the star-triangle transformation is used to establish a correspondence between a nonplanar Ising lattice with two interaction constants and the standard triangular Ising model. The critical point of the nonplanar lattice is obtained exactly for various values of the ratio of the interaction constants. The singularity in the zero-field free energy of the nonplanar lattice is investigated. It is shown to be of the same form as that displayed by the corresponding standard Ising model.
The "star-triangle" and "decoration" transformations are generalized so as to apply to arbitrary mechanical systems coupled to the spins of a standard Ising net. This leads to exact solutions for further plane Ising nets and also for lattices in which the spins on alternate sites have a magnitude greater than S=12S=\frac{1}{2}S=21​. A general class of antiferromagnetic Ising models is constructed; exact closed expressions can be derived for all the thermodynamic and magnetic properties of these models in an arbitrary magnetic field.The magnetizations and susceptibilities of Ising nets in which different spins have different magnetic moments are investigated and a valuable relation between the susceptibilities of the honeycomb and triangular lattices is derived. It is shown how correlation functions involving a given spin can be expressed in terms of correlations involving the nearest-neighbor spins instead.
The three-spin interaction Hamiltonian ${\mathcal{H}}^{(3)}\ensuremath{\equiv}\ensuremath{-}{J}_{3}\ensuremath{\Sigma}{〈\mathrm{ijk}〉}^{}{s}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\sigma}}_{k}\ensuremath{-}H\ensuremath{\Sigma}{i}^{}{s}_{i}\ensuremath{-}{H}^{\ensuremath{'}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\sigma}}_{j}$ (the sss and $\ensuremath{\sigma}$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The ${H}^{\ensuremath{'}}=0$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.xxx are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the ${H}^{\ensuremath{'}}=0$ magnetization is nonzero for all finite TTT, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1\ensuremath{-}\ensuremath{\alpha}$.