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Fortschritte der Physik

The Causal Function in Many Body Problems

Hideki Matsumoto

Published Jan 1, 1977• 99 citations
Publisher

Abstract

Abstract Analytic properties of the causal Green's function in statistical mechanics are investigated. It is shown that the causal function can be obtained as a certain limit of an analytic function. The dual field introduced by Umezawa and Takahashi in their field‐theoretical formulation of statistical mechanics (thermo field dynamics), plays an important role to make the spectral functions real. In a few examples, it is shown explicitly that the Feynman diagram technique is applicable in the thermo‐field dynamics.

DOI
10.1002/prop.19770250101

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Abstract Analytic properties of the causal Green's function in statistical mechanics are investigated. It is shown that the causal function can be obtained as a certain limit of an analytic function. The dual field introduced by Umezawa and Takahashi in their field‐theoretical formulation of statistical mechanics (thermo field dynamics), plays an important role to make the spectral functions real. In a few examples, it is shown explicitly that the Feynman diagram technique is applicable in the thermo‐field dynamics.
摘要 本文从解析性的讨论出发,应用朗道关系改进Дзялошинский的图解法,建立了双时因果格林函数的费曼图解法,它具有正确的零温权限,并可直接用来计算各种非平衡过程的动力学性质。 Abstract Starting from the discussion of analytic properties and using the Landau relation to improve the diagrammatic technique of Dzyaloshinskii, the Feynman diagrammatic technique for double time dependent causal Green's function is established. There is a correct limit of zero temperature and may be used to calculate the various dynamical properties of nonequlibrium processes. 作者及机构信息 章思俊 1. 中国科学院 Authors and contacts CHANG SHIH-CHUN 1. 中国科学院 参考文献 [1] 施引文献
Abstract This paper, on one hand, is an introduction to thermo‐field dynamics (TFD), on the other hand a derivation of some interesting features of Green's functions in the frame of thermo‐field dynamics. It is proved that in the language of TFD, Green's functions at finite temperature take a very simple form. Through the applications of TFD to many‐body systems it is shown that TFD is a straightforward method in dealing with finite temperature many‐body systems. As an application of Green's function method in TFD, we derive a Kogan‐type formula for the non‐equilibrium dynamic conductivity or laser‐pulse heated condensed matters, where electrons and ions are held at different temperatures.
Causality is a fundamental requirement for physical consistency in many-body approximations. In the context of Green's functions, causality requires a positive spectral function and analyticity in the upper half of the complex-frequency plane. These properties are not guaranteed by approximate methods, and formal proofs of causality for the underlying approximation schemes are often difficult to establish. The recently rediscovered Nevanlinna--Pick criterion provides a practical way to test causality directly from Matsubara-frequency data, without performing analytic continuation to the real-frequency axis. In this work, we use the Nevanlinna--Pick criterion to investigate the causality of the ladder dual-fermion approach, a diagrammatic extension of dynamical mean-field theory designed to incorporate nonlocal correlations. The analysis is performed for the spinless Falicov--Kimball model, which provides a useful testing ground because several quantities can be computed exactly or semi-analytically, avoiding Monte Carlo noise. This is particularly important since the Nevanlinna--Pick criterion is highly sensitive to noise in the input data. In addition, we derive several analytical properties of the Pick matrix, providing insight into the structure of the criterion when applied to Green's functions. For small systems, apparent causality violations are observed, but these are found to be associated with finite-size effects. For larger system sizes, no robust signs of causality violations are found, neither within the full ladder dual-fermion approach nor within the leading dual-fermion correction.
S>A formulation of the thermodynamic perturbation theory is proposed which permits the application of quantum field theory methods to quantum statistics at finite temperatures. The method is an extension of the Matsubara technique and is based on expansion of the Green's functions in Fourier series in an imaginary time variable. The technique thus arrived at differs from the usual diagram technique for T = 0 in that summation over discrete values of the imaginary frequency is substituted for integration over the frequencies. The analytical properties of Fourier components of Green's functions are investigated. It is shown that due to the possibility of analytic continuation for solution of various kinetic and nonstationary problems, it is sufficient to know the corresponding equilibrium Green's functions. (auth)
This chapter examines the energy of the resulting quasi-particle using the Green function technique. The field-theoretic Green function technique developed originally by Feynman, Dyson, and others for quantum electrodynamics finds extensive applications to the many body problems encountered in different branches of physics, for its proven efficacy in dealing with the dynamical effects associated with these problems. The Green function formalism is essentially a systematic time-dependent perturbation theory involving certain quantities called the Green functions or propagators. The energy-dependent Green function, G, provides great many physical information about a system, in particular the energy of the quasi-particles. Puff and Whitfield have considered the thermodynamic Green functions for the problem and found at the end of the calculation the zero-temperature limit. In the zero-temperature limit the approximate thermodynamic Green function of Mahan becomes equivalent to the intermediate coupling polaron model of Haeringen and is thus expected to give the same result as given by method of Haeringen.