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Canadian Journal of Physics

Dynamical perturbation theory for the evaluation of the relaxation function

I. M. Kim, Bae‐Yeun Ha

Published Jan 1, 1989• 10 citations
Publisher

Abstract

Use of the infinite continued-fraction representation has played an important role in the evaluation of the relaxation function of many-body systems. By using the method of recurrence relations, we propose here a dynamical perturbation theory for the evaluation of the infinite continued fraction. The analytic expression of the relaxation function is given in terms of the known relaxation function for the unperturbed system. Two illustrative examples are also given.

DOI
10.1139/p89-003

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Use of the infinite continued-fraction representation has played an important role in the evaluation of the relaxation function of many-body systems. By using the method of recurrence relations, we propose here a dynamical perturbation theory for the evaluation of the infinite continued fraction. The analytic expression of the relaxation function is given in terms of the known relaxation function for the unperturbed system. Two illustrative examples are also given.
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.
An established formal way to solve the Liouville equation for dynamical systems with hermitian Hamiltonians is to frame the problem via parametrization in a canonical ensemble. The goal is to solve for the dynamics in the canonical ensemble. This approach allows one to examine temperature-dependent relaxation functions as a part of the construction of the full solution to the Liouville equation. Of course, a full solution to the Liouville equation may not always be necessary as it is the relaxation function which often tends to be of interest. The formalism involves evaluation of infinite continued fractions and this can at times be an especially challenging problem. Here we explore the relaxation function in near-equilibrium systems in the presence of strong nonlinearity. Due to computational challenges, we consider small systems. However, even our small system considerations may be adequate to infer about the relaxation in large systems. We show that the nonlinear systems tend to lead to nonconvergent infinite continued fractions. These continued fractions cannot be approximated perturbatively. We close by presenting a new strategy to evaluate them, thereby demonstrating a successful way to examine relaxation processes in strongly nonlinear systems.
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.
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. 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 numerically exact diagonalization 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.