Local Dynamics in an Infinite Harmonic Chain

Lee, M. Howard

doi:10.3390/sym8040022

“…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

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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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“…In a broadly phrased description, ergodic theorems equate time and ensemble averages [1][2][3][4][5][6], and take various forms both in the precise definition of the term "equate" (which is usually done in the senses of Set Theory, for example, References [7,8]) and of the variety of items whose averages are considered. As regards the former, physical applications of ergodicity (as in References [9,10]) typically steer clear of the niceties (e.g., the qualification of "almost all") of set-theoretical precision [11,12], as is being done here. The varieties of theorems are features of both developments in the course of times and at a particular time of the subject matters which are at interest.…”

Section: Introductionmentioning

confidence: 92%

Ergodic Tendencies in Sub-Systems Coupled to Finite Reservoirs—Classical and Quantal

Englman

¹

2020

Symmetry

0

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Whereas ergodic theories relate to limiting cases of infinite thermal reservoirs and infinitely long times, some ergodicity tendencies may appear also for finite reservoirs and time durations. These tendencies are here explored and found to exist, but only for extremely long times and very soft ergodic criteria. “Weak ergodicity breaking” is obviated by a judicious time-weighting, as found in a previous work [Found. Phys. (2015) 45: 673–690]. The treatment is based on an N-oscillator (classical) and an N-spin (quantal) model. The showing of ergodicity is facilitated by pictorial presentations.

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Local Dynamics in an Infinite Harmonic Chain

Cited by 12 publications

References 43 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

Ergodic Tendencies in Sub-Systems Coupled to Finite Reservoirs—Classical and Quantal

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