Keldysh technique and non-linear sigma-model: basic principles and applications

TL;DR

The article surveys the Keldysh formalism and its functional-integral realization, emphasizing the non-linear sigma-model as a powerful tool for disordered metals and superconductors. It develops the bosonic and fermionic sectors, derives kinetic equations and self-energies, and demonstrates how nonequilibrium transport, noise, and full counting statistics can be computed without recourse to equilibrium analyticity. A central thread is the construction and use of the matrix Q field on the Keldysh contour, its saddle points, and soft modes (diffusons) to capture diffusion, interaction corrections (Altshuler–Aronov, ZBA), and superconducting fluctuations (Usadel, TDGL). The framework unifies disorder, interactions, and superconductivity, enabling systematic derivations of transport coefficients, spectral statistics, and fluctuation phenomena across mesoscopic and nanoscale systems, including drag effects and quantum noise. Overall, it provides a coherent, self-contained toolkit for real-time quantum many-body problems far from equilibrium with broad applicability to condensed-m matter systems.

Abstract

The purpose of this review is to provide a comprehensive pedagogical introduction into Keldysh technique for interacting out-of-equilibrium fermionic and bosonic systems. The emphasis is placed on a functional integral representation of underlying microscopic models. A large part of the review is devoted to derivation and applications of the non-linear sigma-model for disordered metals and superconductors. We discuss such topics as transport properties, mesoscopic effects, counting statistics, interaction corrections, kinetic equation, etc. The sections devoted to disordered superconductors include Usadel equation, fluctuation corrections, time-dependent Ginzburg-Landau theory, proximity and Josephson effects, etc. (This review is a substantial extension of arXiv:cond-mat/0412296.)

Figures (19)

• Figure 1: The closed time contour $\mathcal{C}$. Dots on the forward and the backward branches of the contour denote discrete time points.
• Figure 2: Graphic representation of $G^R$, $G^A$, and $G^K$. The full line represents the classical field component $\phi^{cl}$, while the dashed line the quantum component $\phi^{q}$.
• Figure 3: Graphic representation of the two interaction vertices of the $|\phi|^4$ theory. There are also two complex conjugated vertices with a reversed direction of all arrows.
• Figure 4: Graphic representation of the two interaction vertices of the $\phi^3$ theory. Note the relative factor of one--third between them.
• Figure 5: Self--energy diagrams for the $\phi^3$ theory.