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Brazilian Journal of Physics

Dynamical behavior of the four-body transverse Ising model with random bonds and fields

B. Boechat, Claudette E. Cordeiro, O. F. de Alcantara Bonfim, J. Florencio, F. C. Sá Barreto

Published Dec 1, 2000• 8 citations
Publisher

Abstract

We study the effect of random bonds and fields on the dynamical behavior of the one-dimensional transverse Ising model with four-spin interactions. We consider finite chains of increasing size to determine the time-dependent correlation function and the longitudinal relaxation function of the infinite chain. In this fully disordered system we observe a crossover from a collective modetype of dynamics to that of a central regime.

DOI
10.1590/s0103-97332000000400010

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We study the time dependent behavior of the four-spin interactions Ising chain in the presence of a disordered transverse magnetic field. In order to analyze the effects of field randomness in the dynamical behavior of the system we calculate the time-dependent correlation function and the longitudinal relaxation function of the infinite chain. We consider a bi-modal distribution for the field energy strength, where one of its values equals the exchange coupling energy while the other value can be lower or higher than the coupling energy. We find that an increasing concentration of weaker fields produces an enhancement of the central mode behavior. However, when the concentration of stronger fields increases, the system undergoes a crossover from central mode to collective-type dynamics.
We study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-1/2 transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.
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.
The effects of Gaussian disorder on the dynamics of the one-dimensional spin-1/2 random transverse Ising model are studied in the high-temperature limit by the recursion method. Both spin autocorrelation functions and the corresponding spectral densities are calculated for three types of disordered cases. It is found that when the standard deviation ${\ensuremath{\sigma}}_{J}$ of the exchange coupling (or the standard deviation ${\ensuremath{\sigma}}_{B}$ of the random transverse field) is small, the dynamics of the system undergoes two crossovers in sequence as the mean value of the exchange coupling (or the random transverse field) varies: from a central-peak behavior to a collective-mode one, and then to the precession of free spins in an external transverse magnetic field. However, when ${\ensuremath{\sigma}}_{J}$ or ${\ensuremath{\sigma}}_{B}$ are large enough, there is no crossover, i.e., the dynamics of the system shows a central-peak behavior and a disordered behavior, respectively.