Comparing the Mori formalism and the green function methods

Abstract

Using a simple model described by a Hamiltonian of fermions coupled to bosons, we show that the relaxation function calculated via a low temperature approximation to the Mori memory function is similar, at least to lowest order, to the relaxation function calculated using a Green function formalism.The analytical calculation of dynamic properties of many-body systems is, in general, a difficult task to accomplish and most of the theoretical models that have been proposed to take account of this job involve -in despite of the high degree of complexity of the calculations usually required -some approximations and/or are restricted to a limited range of temperatures or wave-vectors. One of the most useful methods for studying the dynamics of a many-particle system is the memory function method proposed by Mori [1] in 1965. That method is based on a generalized Langevin equation, and is a generalization of the projection-operator technique proposed by Zwanzig [2]. Mori's method is physically appealing because it can show how two different time scales, slow and fast, can possibly arise from Hamiltonian systems and how transport coefficients can be related to the interaction energy [1,3]. The method leads to an expression for the Laplace transform of the autocorrelation function in the form of a continued fraction involving static and dynamical quantities. One of the drawbacks of the method is that the related static properties are not calculated as part of the whole procedure but, rather, are assumed to be known from ab initio. However, the main problem of the procedure seems to be the absence of a prescription on how to cut the continued fraction expansion in which the memory function is expanded. Several approximations, or recipes for a reasonable cut, have been proposed in the literature for the calculation of the memory function [3,4], and the relaxation function so obtained has been successfully compared to experimental data [5,6], and to results obtained by using other methods [7]. Although there is a great number of applications of Mori's method, only a few discussions concerning the essence of the method and the validity of the results obtained whit a particular approximation scheme have appeared in the literature [3,8]. A particularly useful approximation to the method, in the low temperature regime, was introduced by some authors [9,10] almost two decades ago; we will refer to this scheme as the low temperature approximation of Mori's method (LTMM). In the LTMM, the memory function, formulated as a relaxation function, is decoupled in the mode-mode approximation. One of the interesting aspects of the approximation is the fact that it yields all frequency moments exact up to first order in temperature.As it is well known, another very important method to treat low temperature quantum models is the perturbation theory in terms of Green functions [11]. Obviously, each of these two theoretical models, LTMM and Green functions, has its own advantages and drawbacks, and, also, the choice of which method shall be ...

Spin dynamics of the frustrated integer spin antiferromagnetic Heisenberg chain

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“…In the present work, we calculate the relaxation function for the quantum one-dimensional integer spin S Heisenberg antiferromagnet taking into account the magnon-phonon coupling using the memory function method, following a procedure proposed originally by Reiter [8] and further developed by other authors [9]. This method has proven successful in the study of the classical and quantum Heisenberg models in one [10] and two dimensions [11,12], showing good agreement with experimental data, molecular dynamic simulations, and, also, with other theories [13]. For the purpose of numerical calculations we will take S = 1.…”

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Memory function approach to correlated electron transport: A comprehensive review

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