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Physical Review B

Effects of double-Gaussian disorder on the dynamics of the quantum XY chain

Zhen-Bo Xu, Xiang-Mu Kong, Zhongqiang Liu

Published May 14, 2008• 10 citations
Publisher

Abstract

We investigate the dynamics of the random one-dimensional spin-1/2 quantum isotropic $XY$ 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.

DOI
10.1103/physrevb.77.184414

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In this paper, the effects of random variables on the dynamics of the s = 1/2 XY model with the Dzyaloshinskii-Moriya interaction are studied. By means of the recurrence relation method in the high-temperature limit, we calculate the spin autocorrelation functions as well as the corresponding spectral densities for the cases that the exchange couplings between spins or external magnetic fields satisfy the double-Gaussian distribution. It is found that when the standard deviation of random exchange coupling δ J (or the standard deviation of random external field δ B ) is small, the dynamics of the system undergoes a crossover from a collective-mode behavior to a central-peak one. However, when δ J (or δ B ) is large, the crossover vanishes, and the system shows a central-peak behavior or the most disordered one. We also analyze the cases in which the exchange couplings or the external fields satisfy the bimodal and the Gaussian distributions. Our results show that for all the cases considered, the dynamics of the above system is similar to that of the one-dimensional random XY model.
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.
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.
Taking one-dimensional random transverse Ising model (RTIM) with the double-Gaussian disorder for example, we investigated the spin autocorrelation function (SAF) and associated spectral density at high temperature by the recursion method. Based on the first twelve recurrants obtained analytically, we have found strong numerical evidence for the long-time tail in the SAF of a single spin. Numerical results indicate that when the standard deviation σ_{JS} (or σ_{BS}) 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 the standard deviation 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_{JS} or a disordered behavior for large σ_{BS}. In this instance, the long-time tails in the SAFs appear, i.e., C(t)\simt^{-2}. Similar properties are obtained when the random variables (J_{i} or B_{i}) satisfy other distributions such as the double-exponential distribution and the double-uniform distribution.
The dynamics of the one-dimensional spin-1/2 quantum XXZ model with random fields is investigated by the recurrence relations method. When the fields satisfy the bimodal distribution, the system shows a crossover between a collective-mode behavior and a central-peak one with increasing field while the anisotropy parameter Δ is small; a disordered behavior replaces the crossover as Δ increases. For the cases of Gaussian and double-Gaussian distributions, when the standard deviation is small, the results are similar to that of the bimodal distribution; while the standard deviation is large enough, the system only shows a disordered behavior regardless of Δ.