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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Asymptotic behavior of dynamical correlations via perturbative analysis of infinite continued fractions

Surajit Sen, James Christopher Phillips

Published May 1, 1993• 11 citations
Publisher

Abstract

The continued-fraction formalism demonstrates that the Laplace-transformed dynamical correlations can be expressed as infinite continued fractions (ICF's). We propose a generalization of the dynamical convergence method (GDCM) of J. Hong and M. H. Lee [Phys. Rev. Lett. 55, 2375 (1985)] of perturbatively evaluating insoluble ICF's when a closely related ICF is exactly soluble. The proposed method overcomes an existing limitation of the dynamical convergence method which involves accurate computing of ratios of small numbers. The limitations are surmounted by exploiting an ``inversion'' property of ICF's. The GDCM allows computationally fast and simple perturbative evaluation of insoluble ICF's with up to ${10}^{\ensuremath{\xi}}$, \ensuremath{\xi}\ensuremath{\le}6, levels of the insoluble ICF when a related ICF is soluble. The method appears to be appropriate for studies of asymptotic behavior of dynamical correlations described by slowly converging and nonconverging ICF's, which are otherwise insoluble, when closely related soluble ICF's exist. The desirable feature of the GDCM is that the computation times required to solve the ICF's are unrelated to the details of the convergence properties. The method has been applied to recalculate the dynamical-spin pair correlation of a recently studied classical XX spin cluster. This application is described in this work.

DOI
10.1103/physreve.47.3152

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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.

Frontiers in Physics•Nov 26, 2020•11 citations
The continued-fraction formalism demonstrates that the Laplace-transformed dynamical correlations can be expressed as infinite continued fractions (ICF's). We propose a generalization of the dynamical convergence method (GDCM) of J. Hong and M. H. Lee [Phys. Rev. Lett. 55, 2375 (1985)] of perturbatively evaluating insoluble ICF's when a closely related ICF is exactly soluble. The proposed method overcomes an existing limitation of the dynamical convergence method which involves accurate computing of ratios of small numbers. The limitations are surmounted by exploiting an ``inversion'' property of ICF's. The GDCM allows computationally fast and simple perturbative evaluation of insoluble ICF's with up to 10\ensuremathξ{10}^{\ensuremath{\xi}}10\ensuremathξ, \ensuremath{\xi}\ensuremath{\le}6, levels of the insoluble ICF when a related ICF is soluble. The method appears to be appropriate for studies of asymptotic behavior of dynamical correlations described by slowly converging and nonconverging ICF's, which are otherwise insoluble, when closely related soluble ICF's exist. The desirable feature of the GDCM is that the computation times required to solve the ICF's are unrelated to the details of the convergence properties. The method has been applied to recalculate the dynamical-spin pair correlation of a recently studied classical XX spin cluster. This application is described in this work.
The Mori-Lee formalism for solving the Liouville (or Heisenberg) equation of motion for Hermitian systems demonstrates that the Laplace-transformed dynamical correlations in canonical ensembles can be written as (in)finite continued fractions. We show that a model-independent direct summation method allows accurate numerical evaluation of all known classes of these (in)finite continued fractions that arise in dynamics problems and thus provides a powerful technique to study the dynamics of many-body and few-body systems. Some studies on dynamical correlations in s=1/2 quantum spin chains are cited as applications of the method presented.
Mori theory leads to in(finite) continued fractions [I(F)CF's] which upon inverse Laplace transformation (ILT) give the dynamical correlations in Hamiltonian systems. We propose a direct summation method to evaluate these ICF's, e.g., 1/[z+${\mathrm{\ensuremath{\Delta}}}_{1}$/(z+... \ensuremath{\infty})], by replacing them with FCF's with poles L=${10}^{\mathrm{\ensuremath{\zeta}}}$, 2\ensuremath{\le}\ensuremath{\zeta}\ensuremath{\le}5, for ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\mu}}}$=${\mathrm{\ensuremath{\mu}}}^{\mathrm{\ensuremath{\varphi}}}$, \ensuremath{\varphi}2. Long-time dynamics is obtained upon an ILT of the ICF for 0\ensuremath{\le}t\ensuremath{\le}\ensuremath{\tau} with \ensuremath{\tau}=f(\ensuremath{\varphi},\ensuremath{\zeta}) being large. Our studies on dynamical correlations for boundary spins in S=1/2 XY chains agree very well with a recent exact solution for these correlations.
Two representation of the Laplace transforms of time autocorrelation functions, namely the moment expansion and Mori's continued fraction representation, are studied from the point of view of their convergence, with the aim of obtaining new general properties of time autocorrelation functions. First, the relation between them is established by using mathematical techniques in the case of a general dynamical variable, and a direct method, useful for applications, in the case of Hermitian variables. The mathematical structure associated to Mori's generalized random forces is investigated and it is shown that these random forces can be obtained by a Schmidt orthogonalization of the sequence of initial time derivatives of the dynamical variable considered. Then, the convergence criteria for both representations are examined and an illustration is given with the exactly solvable case of an isotopic impurity in a linear chain of coupled harmonic oscillators. Finally, the question of knowing whether the continued fraction expansion is convergent for any Harmitian dynamical variable and any system is discussed, with its implications for the general behaviour of time autocorrelation functions.
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.