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

Continued fraction formalism of linear dynamic conductivity by a combined projection technique

Joung Young Sug, Nam Lyong Kang, Jai Yon Ryu, Sang Don Choi

Published Feb 1, 1995• 6 citations
Publisher

Abstract

Starting with the Liouville equation and with use of the combined projection technique presented earlier [J. Y. Sug, C. H. Choi, and S. D. Choi, Nuovo Cimento B 109, 177 (1994)], we obtain the ensemble average of current of many-electron systems in the presence of oscillatory electric fields. For quite weak fields the dynamic conductivity is given in a continued fraction form. The present form contains the first- and second-order correction parts, while the Mori type contains only the first-order part.

DOI
10.1103/physreve.51.929

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Starting with the Liouville equation and with use of the combined projection technique presented earlier [J. Y. Sug, C. H. Choi, and S. D. Choi, Nuovo Cimento B 109, 177 (1994)], we obtain the ensemble average of current of many-electron systems in the presence of oscillatory electric fields. For quite weak fields the dynamic conductivity is given in a continued fraction form. The present form contains the first- and second-order correction parts, while the Mori type contains only the first-order part.
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.
Three decades ago, the quantum Langevin-like equation (Mori formula) was used to reformulate the linear response theory (Kubo formula) in the continued fraction formalism. Since then, the Mori formula has often been used to evaluate transport coefficients, but only within the Born approximation. Very recently, a rigorous closed-form solution to the Mori formula for transport coefficients has been obtained. In the present study, we prove the equivalence of the closed-form solution to the Mori formula and the Kubo formula using the general relations between correlation functions and then comment on the Born approximation.
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.
Taking the viewpoint that a complete solution of the Liouville equation contains much more information than is necessary in response theory (wherein one is concerned with a particular response), projection techniques are employed to introduce a shortcut in the problem. The treatment supplements and completes the formalism initiated in a previous publication and yields a general equation which is more relevant to response theory in the usual sense. The most natural application is to the theory of transport. The general transport equation is analyzed in the context of electrical conduction. Meaningful similarities to the simple classical Drude model of conduction are exhibited; the relationship of this exact formalism to the Kubo theory of linear response is shown, an exact solution for a step-function stimulus and approximation procedures which go beyond the linear regime are presented, and an expression for the electrical resistivity is explicitly calculated thus making contact with an earlier paper by Kenkre and Dresden. The response equation is also analyzed in the context of electric and magnetic polarization.