Time dependent transverse correlations in the Ising model in D dimensions

Florencio, J.; Sen, Surajit; Lee, M. Howard

doi:10.1590/s0103-97332000000400016

“…Also, from RRII one obtains (da 0 (t)/dt)| 0 0, which precludes a pure time exponential as well as other functions that do not have zero derivative at t 0. The method of recurrence relations have since been applied to a variety of problems, such as the electron gas [33][34][35][36], harmonic oscillator chains [37][38][39][40][41][42][43][44][45][46], many-particle systems [47][48][49][50], spin chains [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66], plasmonic Dirac systems [67,68], dynamics of simple liquids [69,70], etc.…”

Section: The Methods Of Recurrence Relationsmentioning

confidence: 99%

Recent Advances in the Calculation of Dynamical Correlation Functions

Florencio

¹

,

Bonfim

²

2020

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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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“…Also, from RRII one obtains (da 0 (t)/dt)| 0 = 0, which precludes a pure time exponential as well as other functions that do not have zero derivative at t = 0. The method of recurrence relations have since been applied to a variety of problems, such as the electron gas [27,28,30,29], harmonic oscillator chains [31,32,33,34,35,36,37,38,39,40], many-particle systems [44,43,41,42], spin chains [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60], plasmonic Dirac systems [61,62], etc.…”

Section: The Methods Of Recurrence Relationsmentioning

confidence: 99%

Recent advances in the calculation of dynamical correlation functions

Florencio¹,

Bonfim²

2020

Preprint

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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“…In both cases, the ÿelds B i are drawn independently according to bimodal distributions. The main theoretical tool we use is the method of recurrence relations [25][26][27][28][29], which has been successfully used to treat a variety of dynamic problems such as, the velocity autocorrelation function [30], interacting electron gas [31,32], quantum spin systems [18,[33][34][35], classical di usion [36][37][38], simple uids [39], and ergodic theory [40], among others. We should stress that in the present work, we shall use an extension of the method of recurrence relations in order to properly account for the random ÿelds.…”

Section: Introductionmentioning

confidence: 99%

Spin dynamics of the quantum XY chain and ladder in a random field

Nunes

¹

,

Plascak

²

,

Florencio

³

2004

Physica A: Statistical Mechanics and its Applications

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We investigate the Hamiltonian dynamics of two low-dimensional quantum spin systems in a random ÿeld, at the inÿnite-temperature limit: the XY chain and the two-leg XY ladder with interchain Ising interactions. We determine the longitudinal spin autocorrelation functions of the spin-1 2 XY chain and ladder in the presence of disordered ÿelds by using the method of recurrence relations. The ÿrst six basis vectors for the chain and the ÿrst four basis vectors for the ladder of the dynamic Hilbert spaces of z j (t), as well as the corresponding recurrents and moments of the time-dependent autocorrelation function, are analytically computed for bimodal distributions of the ÿelds. We did ÿnd a remarkable result in the disordered models. Cases with a fraction of p sites under ÿeld BB and a fraction of 1 − p sites under the ÿeld BA have the same longitudinal dynamics as those with p sites under ÿeld BA and 1 − p sites under the ÿeld BB. We also ÿnd that both the XY chain and the two-leg XY ladder with Ising interchain coupling in the presence of random ÿelds are sensitive to the percentage of disorder but not to the intensity of the ÿelds.

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Time dependent transverse correlations in the Ising model in D dimensions

Cited by 13 publications

References 1 publication

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

Spin dynamics of the quantum XY chain and ladder in a random field

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