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Physical Review Letters

Critical Index ν of the Vertical-Arrow Correlation Length in the Eight-Vertex Model

James D. Johnson, S. Krinsky, Barry M. McCoy

Published Aug 21, 1972• 27 citations
Publisher

Abstract

We calculate the critical index $\ensuremath{\nu}$ of the correlation length between two vertical arrows in the same column for the eight-vertex model and find $\ensuremath{\nu}=1\ensuremath{-}\frac{\ensuremath{\alpha}}{2}$. We also compute at zero temperature the low-lying excitations of the $X\ensuremath{-}Y\ensuremath{-}Z$ model.

DOI
10.1103/physrevlett.29.492

Related papers

Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the X−Y−Z Hamiltonian

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We consider the correlation function ${G}_{R}$ for two vertical arrows in the same column. We calculate the critical index $\ensuremath{\nu}$ of the correlation length, and we find the scaling relation $\ensuremath{\nu}=1\ensuremath{-}\frac{\ensuremath{\alpha}}{2}$ is satisfied. In the decoupling limit, we prove that the correlation length is not determined by the next-largest eigenvalue. In order to obtain the correct correlation length, it is necessary to integrate over the entire band of complex next-largest eigenvalues. We argue that this is also the situation in the general case of the eight-vertex model. Under certain well-defined assumptions, we compute the correlation length of ${G}_{R}$. We also calculate the low-lying excitation energies of the $X\ensuremath{-}Y\ensuremath{-}Z$ Hamiltonian. In addition to free states existing for $0<\ensuremath{\mu}<\ensuremath{\pi}$, there are bound states appearing in the spectrum for $\ensuremath{\mu}>\frac{\ensuremath{\pi}}{2}$. In the course of our work, we have rewritten the results of Cheng and Wu for the Ising-model correlation functions in an elegant form, using Baxter's elliptic-function parametrization.

Physical Review A•Nov 1, 1973•399 citations

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Arrow-arrow correlations for the six-vertex model

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Spontaneous polarization of the eight-vertex model

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We calculate the critical index \ensuremathν\ensuremath{\nu}\ensuremathν of the correlation length between two vertical arrows in the same column for the eight-vertex model and find \ensuremathν=1\ensuremath−\ensuremathα2\ensuremath{\nu}=1\ensuremath{-}\frac{\ensuremath{\alpha}}{2}\ensuremathν=1\ensuremath−2\ensuremathα​. We also compute at zero temperature the low-lying excitations of the X\ensuremath−Y\ensuremath−ZX\ensuremath{-}Y\ensuremath{-}ZX\ensuremath−Y\ensuremath−Z model.
We consider the correlation function GR{G}_{R}GR​ for two vertical arrows in the same column. We calculate the critical index $\ensuremath{\nu}$ of the correlation length, and we find the scaling relation $\ensuremath{\nu}=1\ensuremath{-}\frac{\ensuremath{\alpha}}{2}$ is satisfied. In the decoupling limit, we prove that the correlation length is not determined by the next-largest eigenvalue. In order to obtain the correct correlation length, it is necessary to integrate over the entire band of complex next-largest eigenvalues. We argue that this is also the situation in the general case of the eight-vertex model. Under certain well-defined assumptions, we compute the correlation length of GR{G}_{R}GR​. We also calculate the low-lying excitation energies of the $X\ensuremath{-}Y\ensuremath{-}Z$ Hamiltonian. In addition to free states existing for $0<\ensuremath{\mu}<\ensuremath{\pi}$, there are bound states appearing in the spectrum for $\ensuremath{\mu}>\frac{\ensuremath{\pi}}{2}$. In the course of our work, we have rewritten the results of Cheng and Wu for the Ising-model correlation functions in an elegant form, using Baxter's elliptic-function parametrization.
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.
The six-vertex model on a square lattice is "exactly solvable" because an exact formula for the free energy can be obtained by the Bethe ansatz. However, exact formulas for the correlations of local bulk observables, such as the orientation of the arrow at a given edge, are, in general, not available. In this Rapid Communication, we consider the isotropic "zero-field" six-vertex model at small Δ. We derive the long-distance asymptotic formula of arrow-arrow correlations, which display power law decays with one anomalous exponent. Our method is based on an interacting fermion representation of the six-vertex model and does not use any information obtained from the exact solution.