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Physical Review Letters

Critical Exponents of the Eight-Vertex Model in Any Dimension

Bill Sutherland

Published Feb 18, 1974• 4 citations
Publisher

Abstract

Results on the critical temperature of the eight-vertex model in any dimension are combined with the Griffiths inequalities for correlations in Ising ferromagnets to yield relations between critical exponents of eight-vertex models of differing dimensionality. Then Baxter's exact results for the two-dimensional, eight-vertex model are able to yield information on the three-dimensional exponents of these models. As in the planar eight-vertex models, the exponents in all dimensions probably vary continuously with the energy parameters.

DOI
10.1103/physrevlett.32.379

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Results on the critical temperature of the eight-vertex model in any dimension are combined with the Griffiths inequalities for correlations in Ising ferromagnets to yield relations between critical exponents of eight-vertex models of differing dimensionality. Then Baxter's exact results for the two-dimensional, eight-vertex model are able to yield information on the three-dimensional exponents of these models. As in the planar eight-vertex models, the exponents in all dimensions probably vary continuously with the energy parameters.
We give a physical interpretation of the entries of the reflection KKK matrices of Baxter's eight-vertex model in terms of an Ising interaction at an open boundary. Although the model still defies an exact solution, we nevertheless obtain the exact surface free energy from a crossing-unitarity relation. The singular part of the surface energy is described by the critical exponents ${\ensuremath{\alpha}}_{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremath{-}\ensuremath{\pi}/2\ensuremath{\mu}$ and ${\ensuremath{\alpha}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1\ensuremath{-}\ensuremath{\pi}/\ensuremath{\mu}$, where \ensuremath{\mu} controls the strength of the four-spin interaction. These values reduce to the known Ising exponents at the decoupling point $\ensuremath{\mu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}/2$ and confirm the scaling relations ${\ensuremath{\alpha}}_{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{b}+\ensuremath{\nu}$ and ${\ensuremath{\alpha}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{b}\ensuremath{-}1$.
The relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two-dimensional systems as a function of finite size is investigated and generalised in two ways. Correlations more general than those of order-order type are included. A form appropriate for anisotropic systems is proposed. The authors present (a) exact results for the Ising and Gaussian models and (b) numerical results for the symmetric eight-vertex (Baxter), continuous q-state Potts, and continuous N-component cubic models.