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 $\nu$ of the correlation length, and we find the scaling relation $\nu =1-\frac{\alpha }{2}$ 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 $X-Y-Z$ Hamiltonian. In addition to free states existing for $0<\mathrm{}\mu <\pi$, there are bound states appearing in the spectrum for $\mu >\frac{\pi }{2}$. 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.