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Physica Scripta

Method of Recurrence Relations and Applications to Many-Body Systems

M. Howard Lee, J. M. Hong, J. Florencio

Published Jan 1, 1987• 57 citations
Publisher

Abstract

The method of recurrence relations allows one to calculate time dependent correlation functions from first principles. It originates from the Kubo scalar product which realizes abstract Hilbert space of dynamical variables. For this realized space, we show that there exists a unique orthogonalization process by recurrence relation. The method of recurrence relations exploits geometric properties of the structure of the realized Hilbert space such as the dimensionality and shape. These geometric properties depend on models as well as static constraints, e.g., temperature, wave vector, size. This method has been applied to several physical models including the homogeneous dense electron gas, the spin-1/2 XY and transverse Ising models, the spin van der Waals model and others. For these models, we have obtained the relaxation function, response function and memory function.

DOI
10.1088/0031-8949/1987/t19b/029

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The method of recurrence relations allows one to calculate time dependent correlation functions from first principles. It originates from the Kubo scalar product which realizes abstract Hilbert space of dynamical variables. For this realized space, we show that there exists a unique orthogonalization process by recurrence relation. The method of recurrence relations exploits geometric properties of the structure of the realized Hilbert space such as the dimensionality and shape. These geometric properties depend on models as well as static constraints, e.g., temperature, wave vector, size. This method has been applied to several physical models including the homogeneous dense electron gas, the spin-1/2 XY and transverse Ising models, the spin van der Waals model and others. For these models, we have obtained the relaxation function, response function and memory function.
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.
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.
Using the recently developed method of recurrence relations, we have studied the time- and frequency-dependent behavior of the two-dimensional electronic system at long wavelengths. Several families of autocorrelation functions including the relaxation and memory functions are obtained exactly. In addition, other linear-response quantities, e.g., the density-density response function and dynamic structure factor, are obtained. This work contains a detailed application of the method of recurrence relations.
This document investigates the dynamics of the one-dimensional S=1/2 isotropic XY model and transverse Ising model in the high temperature limit by using the method of recurrence relations. The authors obtain the relaxation functions as well as some Brownian analogs of a generalized Langevin equation for a tagged spin S x over j in these models, namely, the memory functions and the random forces. It is found that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale. Based on the infinite dimensionality of these Hilbert spaces it is also concluded that S x over j is ergodic in both models.
We investigate the dynamics of the one-dimensional S=(1/2) isotropic XY model and transverse Ising model in the high-temperature limit by using the method of recurrence relations. We obtain the relaxation functions as well as some Brownian analogs of the generalized Langevin equation for a tagged spin Sjx{S}_{j}^{x}Sjx​ in these models, namely, the memory functions and the random forces. We find that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale.