Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of theX−Y−ZHamiltonian

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.