Physical Review Letters

Some Exact Results for the Electric Correlation Functions of the Eight-Vertex Model

Lee-Fen Ko, Barry M. McCoy

Published Jun 16, 1986• 5 citations

Abstract

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 $T<{T}_{c}$ and $T>{T}_{c}$ are considered. In both cases the result is remarkably simple. For $T>{T}_{c}$, 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 $\mathrm{XYZ}$ spin chain are also given.

DOI: 10.1103/physrevlett.56.2645

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Physical Review Letters

Some Exact Results for the Electric Correlation Functions of the Eight-Vertex Model

Lee-Fen Ko, Barry M. McCoy

Published Jun 16, 1986• 5 citations

Publisher

Abstract

T<{T}_{c}

and

T>{T}_{c}

are considered. In both cases the result is remarkably simple. For

T>{T}_{c}

, 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

\mathrm{XYZ}

spin chain are also given.

DOI: 10.1103/physrevlett.56.2645

Related papers

Perturbation calculation around the two-dimensional Ising model

Lee-Fen Ko

Physical review. B, Condensed matterJun 1, 19881 citation

Electric correlation function for the eight-vertex model for T→T−c to first order in the four-spin coupling

Lee-Fen Ko, Barry M. McCoy

Physics Letters AOct 1, 19864 citations

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

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.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.

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The eight-vertex and XYZ model

Barry M. McCoy

Dec 1, 20090 citations

T<{T}_{c}

and

T>{T}_{c}

are considered. In both cases the result is remarkably simple. For

T>{T}_{c}

, 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

\mathrm{XYZ}

spin chain are also given.

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.

F

model and that of the four-state Potts model.

XY

model in one dimension. In higher dimensions, series are calculated for the triangular, square, and simple cubic lattices, and numerical estimates for the critical exponents are found to agree with expectations, within errors.