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

Florencio,Joao , Jr, M H Lee

Published Nov 5, 1986• 0 citations
OpenAlex

Abstract

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.

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We investigate 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. We obtain the relaxation functions as well as some Brownian analogs of the generalized Langevin equation for a tagged spin Sjx{S}_{j}^{x}Sjx​ in these models, namely, the memory functions and the random forces. We find 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.
We investigate the dynamics of the random one-dimensional spin-1/2 quantum isotropic XYXYXY model in high-temperature limit by the recurrence relations method. Both the time-dependent transverse autocorrelation function and the corresponding spectral density are calculated for the cases that the exchange couplings between spins or transverse fields satisfy the double-Gaussian distribution. It is found that when the value of ${\ensuremath{\sigma}}_{J}$ (standard deviation of random exchange coupling) or ${\ensuremath{\sigma}}_{B}$ (standard deviation of random external magnetic field) is small, the dynamics of the system undergoes a crossover between a central-peak behavior and a collective-mode one. However, when ${\ensuremath{\sigma}}_{J}$ or ${\ensuremath{\sigma}}_{B}$ is large enough, the crossover vanishes, and the system shows the most disordered behavior or a central-peak behavior. We also investigate the cases that the couplings or fields satisfy the bimodal distribution and Gaussian distribution, which are the two special cases of double-Gaussian distribution, and find that the dynamics of the system is similar to that of the one-dimensional transverse Ising system.
The behavior at finite temperatures of the dynamical transverse autocorrelation of a spin is investigated for two special cases of the one-dimensional XY\mathrm{XY}XY model. We find for temperatures restricted by $\frac{J}{\mathrm{kT}}>{e}^{\ensuremath{-}2\mathrm{Jt}}$ that its large-time behavior is given by $〈{S}_{i}^{x}(t){S}_{i}^{x}〉\ensuremath{\sim}C(\mathrm{kT})\mathrm{exp}[\ensuremath{-}\frac{1}{2}{J}^{2}{t}^{2}+\frac{2{J}^{2}{t}^{2}}{\mathrm{ln}(\mathrm{Jt})}]$ for the transverse Ising model with field equal to exchange constant and by its square for the isotropic XY\mathrm{XY}XY model without field.
The dynamics of the two-dimensional spin-12\frac{1}{2}21​ random transverse Ising model is studied by means of the recurrence relations method. Both the autocorrelation functions and the corresponding spectral densities are calculated in the cases that the exchange couplings or fields satisfy three typical distributions, respectively. For the cases of Gaussian and double-Gaussian distribution, when the standard deviation of the random variable is small, the dynamics of the system undergoes a crossover from a central-peak behavior to a collective-mode one, the result is similar to that of the bimodal distribution. While the standard deviation is large enough, the crossover disappears and the dynamics of the system shows a central-peak behavior or a disordered behavior.
The nonthermal quantum relaxation of the magnetization under nonequilibrium initial conditions is studied for the transverse Ising chain and for the one-dimensional isotropic XY model. In the absence of a heat bath the transverse Ising system exhibits an oscillating relaxation with a decaying amplitude. At the critical transverse field some stationary properties become independent of the initial state. The isotropic XY chain with a spatially inhomogeneous initial state shows a slow, modulated algebraic decay of the magnetization. Analysis of the two-time autocorrelation function reveals the phenomenon of aging.