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Physical review. B, Condensed matter

Dynamics of the random one-dimensional transverse Ising model

J. Florencio, F. C. Sá Barreto

Published Oct 1, 1999• 34 citations
Publisher

Abstract

We study the dynamics of the spin-1/2 random transverse Ising model in the high-temperature limit by means of the method of recurrence relations. We analyze two types of disorder: a disorder on the transverse field, and a disorder on the exchange coupling. We find that the dynamics undergoes a crossover from a central peak behavior onto a collective mode behavior as a function of the disorder.

DOI
10.1103/physrevb.60.9555

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We study the dynamics of the spin-1/2 random transverse Ising model in the high-temperature limit by means of the method of recurrence relations. We analyze two types of disorder: a disorder on the transverse field, and a disorder on the exchange coupling. We find that the dynamics undergoes a crossover from a central peak behavior onto a collective mode behavior as a function of the disorder.
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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 dynamical properties of one-dimensional random transverse Ising model (RTIM) with a double-Gaussian disorder is investigated by the recursion method. Based on the first twelve recurrences derived analytically, the spin autocorrelation function (SAF) and associated spectral density at high temperature were obtained numerically. Our results indicate that when the standard deviation σ J (or σ B ) of the exchange couplings J i (or the random transverse fields B i ) is small, no long-time tail appears in the SAF. The spin system undergoes a crossover from a central-peak behavior to a collective-mode behavior, which is the dynamical characteristics of RTIM with the bimodal disorder. However, when σ J (or σ B ) is large enough, the system exhibits similar dynamics behaviors to those of the RTIM with the Gaussian disorder, i.e., the system exhibits an enhanced central-peak behavior for large σ J or a disordered behavior for large σ B . In this instance, SAFs exhibit a similar long-time tail, i.e., C ( t ) ∼ t −2 for large t . Similar properties are obtained when J i (or B i ) satisfy the double-exponential distribution or the double-uniform distribution. Besides, when both the standard deviations and the mean values of the exchange couplings are small, the effects of the Gaussian random bonds may drive the system undergo two crossovers from a triplet state to a doublet state, and then to a collective-mode state.
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.
We study the time evolution of the one-dimensional random-bond transverse Ising model with four-spin interactions. We calculate the time-dependent correlation function as well as the longitudinal relaxation function of the infinite chain. We analyze how the presence of disorder affect the dynamical behavior of the system in comparison with the pure model. We find that the main effect of disorder is to produce a crossover from a central mode to a collective-mode type of dynamics, as the concentration of weaker bonds is enhanced. Such crossover is also present in the case of an increase in bond dilution.
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