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Physical review. B, Condensed matter

Generalized Gorkov’s equation for nonequilibrium superconductivity

Takeshi J. Inagaki

Published Jul 1, 1995• 2 citations
Publisher

Abstract

Making use of the inversion method, which is invented to discuss the characteristics of phase transition in a perturbative manner, a fundamental equation for nonequilibrium superconductivity is derived. In the course of the calculation, the generalized contour-ordered Green's-function formalism is used to treat the initial time correlation function and real-time Green's function in a unified manner. The result is a set of self-consistent integral equations and the Gorkov's equation is given a posteriori as one of these components describing the initial time correlation. By virtue of the contour-ordered Green's-function formalism, it is not necessary to employ any assumption to consider the effect of the initial time correlation on the system at subsequent time. Owing to this nature, our result helps in discussing transient behavior of the system induced by a supercritical current that appears in some kinds of superconducting device.

DOI
10.1103/physrevb.52.1219

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Making use of the inversion method, which is invented to discuss the characteristics of phase transition in a perturbative manner, a fundamental equation for nonequilibrium superconductivity is derived. In the course of the calculation, the generalized contour-ordered Green's-function formalism is used to treat the initial time correlation function and real-time Green's function in a unified manner. The result is a set of self-consistent integral equations and the Gorkov's equation is given a posteriori as one of these components describing the initial time correlation. By virtue of the contour-ordered Green's-function formalism, it is not necessary to employ any assumption to consider the effect of the initial time correlation on the system at subsequent time. Owing to this nature, our result helps in discussing transient behavior of the system induced by a supercritical current that appears in some kinds of superconducting device.
We develop a time-local nonequilibrium Green's function formulation for real-time dynamics in quantum systems coupled to superconducting leads. The superconducting lead self-energy is a strongly frequency-dependent matrix in Nambu space, giving rise to nonlocal memory kernels in the time domain. This makes direct propagation of the Kadanoff-Baym (KB) equations computationally demanding. To overcome this difficulty, we extend the auxiliary-mode expansion, originally developed for normal-metal leads, to Nambu-space self-energies. This allows us to decompose the superconducting lead self-energy into a finite number of exponential modes and to transform the KB equations with memory integrals into a closed set of ordinary differential equations. The resulting time-local equations enable efficient real-time simulations under general time-dependent bias voltages, superconducting phases, and one-body Hamiltonians of the central system, while retaining the memory effects induced by superconducting leads. As an application, we analyze voltage-quench dynamics in a superconductor-quantum-dot-superconductor junction and show that, after a dc bias is suddenly applied, the system evolves through a transient regime and relaxes to an ac Josephson periodic steady state. The resulting periodic steady-state current agrees with the Floquet Green's function solution, validating the present real-time formulation.
We present a unified framework for equilibrium and nonequilibrium many-body perturbation theory. The most general nonequilibrium many-body theory valid for general initial states is based on a time-contour originally introduced by Konstantinov and Perel'. The various other well-known formalisms of Keldysh, Matsubara and the zero-temperature formalism are then derived as special cases that arise under different assumptions. We further present a single simple proof of Wick's theorem that is at the same time valid in all these flavors of many-body theory. It arises simply as a solution of the equations of the Martin-Schwinger hierarchy for the noninteracting many-particle Green's function with appropriate boundary conditions. We further discuss a generalized Wick theorem for general initial states on the Keldysh contour and derive how the formalisms based on the Keldysh and Konstantinov-Perel'-contours are related for the case of general initial states.
The Green's function method is one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has proved invaluable in many research fields. This book provides a unique, self-contained introduction to nonequilibrium many-body theory. Starting with basic quantum mechanics, the authors introduce the equilibrium and nonequilibrium Green's function formalisms within a unified framework called the contour formalism. The physical content of the contour Green's functions and the diagrammatic expansions are explained with a focus on the time-dependent aspect. Every result is derived step-by-step, critically discussed and then applied to different physical systems, ranging from molecules and nanostructures to metals and insulators. With an abundance of illustrative examples, this accessible book is ideal for graduate students and researchers who are interested in excited state properties of matter and nonequilibrium physics.
We present a Green's-function formalism generalized to treat arbitrary time-dependent systems initially prepared with an arbitrary density matrix. On the basis of a simultaneous expansion of the usual real-time development operator and the density matrix, we define a Green's function ordered along an extended contour. In this way we are able to treat real-time many-particle Green's functions and initial correlations on the same footing. Both can be decomposed by Wick's theorem. Thus we may utilize familiar diagrammatic analysis for evaluating the Green's function, and finally arrive at a generalized Dyson's equation. The Green's function can be represented by a (3\ifmmode\times\else\texttimes\fi{}3) matrix, which includes the Green's functions used in the Feynman, Matsubara, and Keldysh theories, if the corresponding statistical average is taken. From a matrix representation of Dyson's equation, a basic set of five coupled equations is derived, which explicitly shows the corrections to the Keldysh theory due to an arbitrary initial many-particle density matrix.
By means of Green's functions methods, Gorkov derived the BCS theory on the basis of the Ansatz that the correlation part of the second-order Green's function could be factored in the form χχ* where χ is a two-particle function closely related to the Ginzburg–Landau order-parameter. Since the density matrix is an initial value of a Green's function, Gorkov's Ansatz is equivalent to an assumption about the 2-matrix. The present paper considers circumstances in which the Gorkov Ansatz is exactly satisfied by a system of a definite number of fermions.