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

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

doi:10.1590/s0103-97332000000400010

“…The time correlation function was obtained for one dimensional and infinite temperature [93], where the Gaussian behavior shown in the usual transverse Ising model was ruled out. The effects of disorder on the dynamics of that model were obtained for the cases where the random variables are drawn from bimodal distributions of random couplings and fields [16,94,95]. Dynamical correlation functions were also obtained for the system at finite temperatures, ranging from T 0 to T ∞ [96].…”

Section: Applications To Interacting Systemsmentioning

confidence: 99%

Recent Advances in the Calculation of Dynamical Correlation Functions

Florencio

¹

,

Bonfim

²

2020

Self Cite

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We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.

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“…The time correlation function was obtained for one dimensional and infinite temperature [83], where the Gaussian behavior shown in the usual transverse Ising model was ruled out. The effects of disorder on the dynamics of that model were obtained for the cases where the random variables are drawn from bimodal distributions of random couplings and fields [16,84,85]. Dynamical correlation functions were also obtained for the system at finite temperatures, ranging from T = 0 to T = ∞ [86].…”

Section: Applications To Interacting Systemsmentioning

confidence: 99%

Recent advances in the calculation of dynamical correlation functions

Florencio¹,

Bonfim²

2020

Preprint

Self Cite

0

View full text Add to dashboard Cite

We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.

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“…Only recently, since the disorder effects besides the effects of the temperature has been shown to affect the dynamics behavior of spin systems in a drastic way providing a very rich area of investigation, more and more attention has been paid to the random quantum spin systems, especially on the low-dimensional systems, such as the random transverse Ising model (RTIM), [13][14][15][16][17][18][19][20] XY chain, [21][22][23][24] XX and XXZ chain, [25] one-dimensional Blume-Capel model, [26] etc. The known re-sults indicate that the disorder effects may drastically slow down the behavior of the disordered averaged spin-spin correlation function, [25] including the SAFs.…”

Section: Introductionmentioning

confidence: 99%

“…For example, as the couplings vary, the Gaussian behavior of xx-relaxation of the single impurity in XY chain slows down to the stretched exponential-like relaxation at T = ∞. [27] However, as far as we know, although the short-time behaviors of the SAFs have been reported, [15][16][17][18][19][21][22][23] the long-time behaviors of them are still not understood. An important issue in this work is to explore whether it is possible to realize a situation in which one may obtain the long-time behavior of SAFs of the random transverse Ising chain, where the exchange couplings and the transverse fields are independent random variable satisfying a double-Gaussian distribution.…”

Section: Introductionmentioning

confidence: 99%

The spin dynamics of the random transverse Ising chain with a double-Gaussian disorder

Liu¹,

Jiang²,

Kong³

2014

Chinese Phys. B

2

0

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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 collectivemode 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.

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

Cited by 9 publications

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Recent Advances in the Calculation of Dynamical Correlation Functions

Recent Advances in the Calculation of Dynamical Correlation Functions

Recent advances in the calculation of dynamical correlation functions

The spin dynamics of the random transverse Ising chain with a double-Gaussian disorder

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