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arXiv:cond-mat/9510152 · Physical Review Letters

Surface Critical Phenomena and Scaling in the Eight-Vertex Model

Murray T. Batchelor, Yu-Sen Zhou

Published Jan 1, 1996• 18 citations
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Abstract

We give a physical interpretation of the entries of the reflection $K$ 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 ${\ensuremath{\alpha}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{b}\ensuremath{-}1$.

DOI
10.1103/physrevlett.76.14

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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 \ensuremathαs=2\ensuremath−\ensuremathπ/2\ensuremathμ{\ensuremath{\alpha}}_{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremath{-}\ensuremath{\pi}/2\ensuremath{\mu}\ensuremathαs​=2\ensuremath−\ensuremathπ/2\ensuremathμ and \ensuremathα1=1\ensuremath−\ensuremathπ/\ensuremathμ{\ensuremath{\alpha}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1\ensuremath{-}\ensuremath{\pi}/\ensuremath{\mu}\ensuremathα1​=1\ensuremath−\ensuremathπ/\ensuremathμ, where \ensuremath{\mu} controls the strength of the four-spin interaction. These values reduce to the known Ising exponents at the decoupling point \ensuremathμ=\ensuremathπ/2\ensuremath{\mu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}/2\ensuremathμ=\ensuremathπ/2 and confirm the scaling relations \ensuremathαs=\ensuremathαb+\ensuremathν{\ensuremath{\alpha}}_{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{b}+\ensuremath{\nu}\ensuremathαs​=\ensuremathαb​+\ensuremathν and \ensuremathα1=\ensuremathαb\ensuremath−1{\ensuremath{\alpha}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{b}\ensuremath{-}1\ensuremathα1​=\ensuremathαb​\ensuremath−1.
We obtain the diagonal reflection matrices for a recently introduced family of dilute lattice models in which the model can be viewed as an Ising model in a magnetic field. We calculate the surface free energy from the crossing-unitarity relation and thus directly obtain the critical magnetic surface exponent for L odd and surface specific heat exponent for L even in each of the various regimes. For L = 3 in the appropriate regime we obtain the Ising exponent , which is the first determination of this exponent without the use of scaling relations.
The eight-vertex model solved by Baxter is shown to be equivalent to two Ising models with nearest-neighbor coupling interacting with one another via a four-spin coupling term. The critical properties of the model in the weak-coupling limit are in agreement with the scaling hypothesis. In this limit where $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, the critical indices obey $\frac{\ensuremath{\gamma}}{{\ensuremath{\gamma}}_{0}}=\frac{\ensuremath{\beta}}{{\ensuremath{\beta}}_{0}}=\frac{\ensuremath{\nu}}{{\ensuremath{\nu}}_{0}}=1\ensuremath{-}\frac{1}{2}\ensuremath{\alpha}$, $\frac{\ensuremath{\delta}}{{\ensuremath{\delta}}_{0}}=\frac{\ensuremath{\eta}}{{\ensuremath{\eta}}_{0}}=1$, with the subscripts zero denoting the index values for the ordinary two-dimensional Ising model.
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.