Relaxation Functions, Memory Functions, Random Forces, and Ergodicity in the One-Dimensional Spin-1/2 XY and Transverse Ising Models

摘要

This document investigates the dynamics of the one-dimensional S=1/2 isotropic XY model and transverse Ising model in the high temperature limit by using the method of recurrence relations. The authors obtain the relaxation functions as well as some Brownian analogs of a generalized Langevin equation for a tagged spin S x over j in these models, namely, the memory functions and the random forces. It is found that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale. Based on the infinite dimensionality of these Hilbert spaces it is also concluded that S x over j is ergodic in both models.