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Physical Review A

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

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

Published Nov 1, 1973• 399 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.8.2526

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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.

Physical Review Letters•Feb 18, 1974•4 citations
We consider the correlation function GR{G}_{R}GR​ for two vertical arrows in the same column. We calculate the critical index \ensuremathν\ensuremath{\nu}\ensuremathν of the correlation length, and we find the scaling relation \ensuremathν=1\ensuremath−\ensuremathα2\ensuremath{\nu}=1\ensuremath{-}\frac{\ensuremath{\alpha}}{2}\ensuremathν=1\ensuremath−2\ensuremathα​ 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−ZX\ensuremath{-}Y\ensuremath{-}ZX\ensuremath−Y\ensuremath−Z Hamiltonian. In addition to free states existing for 0<\ensuremathμ<\ensuremathπ0<\ensuremath{\mu}<\ensuremath{\pi}0<\ensuremathμ<\ensuremathπ, there are bound states appearing in the spectrum for \ensuremathμ>\ensuremathπ2\ensuremath{\mu}>\frac{\ensuremath{\pi}}{2}\ensuremathμ>2\ensuremathπ​. 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.
We calculate the two-particle contribution to the spectral weight function of the electric correlation functions of the eight-vertex model in the scaling limit to first order in the four-spin coupling. Both T<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​ are considered. In both cases the result is remarkably simple. For T>TcT>{T}_{c}T>Tc​, the effect of the emergence of the bound state from the two-particle continuum is seen. The corresponding results for the Ashkin-Teller model and the XYZ\mathrm{XYZ}XYZ spin chain are also given.
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.