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arXiv:1711.10758

The Memory Function Formalism: A Review

Komal Kumari, Navinder Singh

Published Nov 29, 2017• 0 citations
arXivHTMLPDF

Abstract

An introduction to the Zwanzig-Mori-Götze-Wölfle memory function formalism (or generalized Drude formalism) is presented. This formalism is used extensively in analyzing the experimentally obtained optical conductivity of strongly correlated systems like cuprates and Iron based superconductors etc. For a broader perspective both the generalised Langevin equation approach and the projection operator approach for the memory function formalism are given. The Götze-Wölfle perturbative expansion of memory function is presented and its application to the computation of the dynamical conductivity of metals is also reviewd. This review of the formalism contains all the mathematical details for pedagogical purposes.

DOI
10.48550/arxiv.1711.10758

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An introduction to the Zwanzig-Mori-Götze-Wölfle memory function formalism (or generalized Drude formalism) is presented. This formalism is used extensively in analyzing the experimentally obtained optical conductivity of strongly correlated systems like cuprates and Iron based superconductors etc. For a broader perspective both the generalised Langevin equation approach and the projection operator approach for the memory function formalism are given. The Götze-Wölfle perturbative expansion of memory function is presented and its application to the computation of the dynamical conductivity of metals is also reviewd. This review of the formalism contains all the mathematical details for pedagogical purposes.
Abstract An introduction to the Zwanzig–Mori–Götze–Wölfle memory function formalism (or generalized Drude formalism) is presented. This formalism is used extensively in analyzing the experimentally obtained optical conductivity of strongly correlated systems such as cuprates and iron-based superconductors. For a broader perspective both the generalized Langevin equation approach and the projection operator approach for the memory function formalism are given. The Götze–Wölfle perturbative expansion of memory function is presented and its application to the computation of the dynamical conductivity of metals is also reviewed. This digest of the formalism contains all the mathematical details for pedagogical purposes.
This chapter analyzes the memory functions in detail. The tool is the projection operator technique introduced by R. Zwanzig and extended by H. Mori. With the help of this technique, the dispersion relations can be made into a powerful and versatile instrument as our later applications, particularly to systems of broken symmetry, should amply demonstrate. However, the projector formalism can be applied to quantum systems as well. In order not to be encumbered by incidentals, the chapter considers the formally simplest case: fluctuations of a single classical dynamical variable. It looks at the spin dynamics, to see how all of the formalism can be used. The chapter discusses the magnetization correlation function and the equilibrium-averaged autocorrelation function. The formal developments were presented for a classical system because such objects as, say, the Liouville operator are a little more familiar classically.
Memory function formalism or projection operator technique is an extremely useful method to study the transport and optical properties of various condensed matter systems. A recent revival of its uses in various correlated electronic systems is being observed. It is being used and discussed in various contexts, ranging from nonequilibrium dynamics to the optical properties of various strongly correlated systems such as high temperature superconductors. However, a detailed discussion on this method, starting from its origin to its present day applications at one place is lacking. In this paper, we attempt a comprehensive review of the memory function approach focusing on its uses in studying the dynamics and the transport properties of correlated electronic systems.
We consider the relationship of the dual Lanczos transformation method, introduced in paper I, to the projection operator methods of Mori and Zwanzig for treating relaxation processes. It is shown that Mori’s memory function formalism and ‘‘generalized’’ Langevin description of relaxation processes represents a special case in the reversible limit of the ‘‘global’’ dual Lanczos transformation method. Unlike Mori’s memory function formalism, the present results are not limited to the description of autocorrelation functions for reversible systems. Our results apply to auto- and cross-correlation functions, as well as, the ensemble average of observables for both reversible and irreversible systems. In fact, we are able to construct a continued fraction representation of both auto- and cross-correlation functions. Introducing a ‘‘bath’’ dual Lanczos transformation scheme, we show that the first bath dual Lanczos projection gives rise to the Zwanzig projection operator method for constructing an equation of motion for a reduced distribution function. Additional bath dual Lanczos projections allow us to go beyond the usual results obtained in applications of the Zwanzig method.
The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.
We introduce a combinatorial version Mori-Zwanzig theory and develop from it a family of self-consistent evolution equations for the correlation function or Green's function of interactive many-body systems. The core idea is to use an ansatz to rewrite the memory kernel (self-energy) of the regular Mori-Zwanzig equation as a function composition of the correlation (Green's) function. Then a series of algebraic combinatorial tools, especially the commutative and noncommutative Bell polynomials, are used to determine the exact Taylor series expansion of the composition function. The resulting combinatorial Mori-Zwanzig equation (CMZE) yields novel non-perturbative expansions of the equation of motion for the correlation (Green's) function. The structural equation for deriving such a combinatorial expansion resembles the combinatorial Dyson-Schwinger equation and may be viewed as its temporal-domain analogue. After introducing the abstract word and tree representation of the CMZE, we show its wide-range application in classical, stochastic, and quantum many-body systems. In all these examples, the new self-consistent expansions we obtained with the CMZE are similar to the diagrammatic skeleton expansions used in quantum many-body theory and lattice statistical field theory. We expect such a new framework can be used to calculate the correlation (Green's) function for strongly correlated/interactive many-body systems.