Analytical expressions for momentum autocorrelation function  of a classic diatomic chain

Yu, Ming B.

doi:10.1140/epjb/e2017-70752-1

“…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%

“…The problem of a mass impurity in the harmonic chain was solved later, and its dynamical correlation functions were found to have the same form as in the quantum electron gas in two dimensions, thus showing that unrelated quantities in these two models displayed the same dynamical behavior, that is, the have dynamic equivalence [76]. It should be mentioned that harmonic oscillator chains have been the subject of a considerable amount of work with the method of recurrence relations [38][39][40][41][42][43][44][45][46].…”

Section: Applications To Interacting Systemsmentioning

confidence: 96%

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%

“…The problem of a mass impurity in the harmonic chain was solved later, and its dynamical correlation functions were found to have the same form as in the quantum electron gas in two dimensions, thus showing that unrelated quantities in these two models displayed the same dynamical behavior, that is, the have dynamic equivalence [66]. It should be mentioned that harmonic oscillator chains have been the subject of a considerable amount of work with the method of recurrence relations [32,33,34,35,36,37,38,39,40].…”

Section: Applications To Interacting Systemsmentioning

confidence: 96%

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.

show abstract

“…The RR method is also used to study oscillator chains, e.g., a monatomic chain without [15] or with a mass impurity [16], a Bethe lattice [17], a diatomic chains without [18] or with a mass impurity [19,20], a monatomic chain with a full impurity, i.e. an impurity with different mass and Hooke constant [21] as well as a diatomic chain with a full impurity [22], etc.…”

Section: Introductionmentioning

confidence: 99%

The Memory Function of an Impurity in Mass and Hooke Constant in a Diatomic Chain

Yu

2023

J Adv Mater Sci Eng

0

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The memory function of an impurity with different mass and Hooke constant in a classical diatomic chain is studied by means of the recurrence relations method. The Laplace transform of the memory function has two pairs of resonant poles and three branch cuts. The poles contribute a cosine and the cuts contribute acoustic and optic branches, which are expressed as a convolution of a difference of two sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of Jacobian elliptic function sn(u) along the real axis in a complex u+iv plane for the acoustic branch, and integrals of nd(v) along a contour parallel to the imaginary axis of the plane for the optical branch, respectively. Different special cases are discussed in detail. It shows that a perfect monatomic or diatomic chain and such a chain with a mass impurity share the same memory function except for a constant factor, and that the pole contribution exists only if the impurity has both different mass and Hooke constant.

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Analytical expressions for momentum autocorrelation function of a classic diatomic chain

Cited by 10 publications

References 23 publications

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 Memory Function of an Impurity in Mass and Hooke Constant in a Diatomic Chain

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