Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Annalen der Physik

Dynamic nonlinear σ‐model of electron localization

M. L. Horbach, Gerd Schön

Published Jan 1, 1993• 33 citations
Publisher

Abstract

Abstract We review the nonlinear σ‐model approach to the localization problem. However, different from usual, we do not make use of replicas or superfields in order to perform the averaging over the realizations of the disorder. Instead, we pursue the derivation of the nonlinear σ‐model as an effective theory for the disordered electron system within the framework of the so‐called time‐path formalism, which is a real‐time finite‐temperature quantum field theory. This formalism is especially suitable for the description of the disordered electron system due to the automatic normalization of real‐time quantities, thereby avoiding the denominator problem. We shall compare this dynamic method with the replica‐ and superfield‐methods as the perturbative level of the calculation of physical quantities.

DOI
10.1002/andp.19935050106

Related papers

Stochastic quantization approach to disordered fermions

E. Kolley, Winfried Kolley

Abstract The method of the stochastic quantization is applied to an electron system described by the random Hubbard model. Two different stochastic calculi are performed corresponding to Euclidean and non‐Euclidean versions of the quantum field theory in terms of Grassmann variables. In averaging over the quenched disorder a nontrivial normalization denominator can be avoided. An averaged, effective two‐time action is given as basis for treating localization.

physica status solidi (b)•Apr 1, 1987•3 citations

The Anderson-Mott transition

D. Belitz, T. R. Kirkpatrick

The interacting disordered electron problem is reviewed with emphasis on the quantum phase transitions that occur in a model system and on the field-theoretic methods used to describe them. An elementary discussion of conservation laws and diffusive dynamics is followed by a detailed derivation of the extended nonlinear sigma model, which serves as an effective field theory for the problem. A general scaling theory of metal-insulator and related transitions is developed, and explicit renormalization-group calculations for the various universality classes are reviewed and compared with experimental results. A discussion of pertinent physical ideas and phenomenological approaches to the metal-insulator transition not contained in the sigma-model approach is given, and phase-transition aspects of related problems, like disordered superconductors and the quantum Hall effect, are discussed. The review concludes with a list of open problems.

Reviews of Modern Physics•Apr 1, 1994•988 citations

New method for studying disordered quantum systems

H. G. Schuster, Vitor Rocha Vieira

The effect of an electron heat bath on the motion of an electron system is studied using a quantum Langevin equation and deriving the influence functional of the Feynman-Vernon approach. The generating functional for the time-dependent, finite-temperature correlation functions of a disordered electron system with interactions is written down, and some of its symmetry properties are analyzed. Since the disorder only affects the time evolution of the system our formalism avoids the replica trick and provides a natural physical framework for studying a large variety of open problems in the field of disordered systems. As an example, we give an explicit expression for the averaged generating functional for a disordered electron system in the presence of a Coulomb interaction in terms of two complex matrix fields.

Physical review. B, Condensed matter•Jul 1, 1986•21 citations

Dynamic Field‐Theoretic Approach to a Disordered Model Superconductor

E. Kolley, Winfried Kolley

Abstract On the basis of the attractive Hubbard model with a Gaussian random potential we describe superconductivity in the presence of quenched disorder. A dynamic path‐integral formalism with an 8‐component Keldysh‐Nambu spinor is shown to be convenient for solving the averaging problem without the recourse to the replica trick. The effective action is bosonized in terms of collective variables, namely local Q‐matrices and real space pair fields. Saddle point values are calculated in two simplified limits.

Annalen der Physik•Jan 1, 1989•0 citations

Schwinger-Keldysh approach to disordered and interacting electron systems: Derivation of Finkelstein’s renormalization-group equations

Claudio Chamon, Andreas W. W. Ludwig, Chetan Nayak

We develop a dynamical approach based on the Schwinger-Keldysh formalism to derive a field-theoretic description of disordered and interacting electron systems. Within this formalism we calculate the perturbative renormalization-group (RG) equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein (Zh. \'Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]) using replicas. The major simplifying feature of this approach, as compared to Finkelstein's, is that instead of $\stackrel{\ensuremath{\rightarrow}}{N}0$ replicas, we only need to consider $N=2$ species. We compare the dynamical Schwinger-Keldysh approach and the replica methods, and we present a simple and pedagogical RG procedure to obtain Finkelstein's RG equations.

arXiv:cond-mat/9810282•Physical review. B, Condensed matter•Jul 15, 1999•65 citations

Path integral approach to quantum thermalization

Alexander Altland, Kun Woo Kim, Tobias Micklitz

We introduce a quasiclassical Green's function approach describing the unitary yet irreversible dynamics of quantum systems effectively acting as their own environment. Combining a variety of concepts of quantum many-body theory, notably the nonlinear σ model of disordered systems, the G Σ formalism for strong correlations, and real-time path integration, the theory is capable of describing a wide range of system classes and disorder models. It extends previous work beyond perturbation theory (in inverse Hilbert-space dimensions), enabling a description of thermalization dynamics from short scattering times, through the onset of ergodicity at an effective “Thouless time,” up to the many-body Heisenberg time. We illustrate the approach with two case studies: (1) a brickwork model of unitarily coupled quantum circuits with and without conserved symmetries and (2) an array of capacitively coupled quantum dots. Using the spectral form factor as a test observable, we find good agreement with numerical simulations. We present our formalism in a self-contained and pedagogical manner, aiming to provide a transferable toolbox for the first-principles description of many-body chaotic quantum systems in regimes of strong entanglement.

arXiv:2509.06028•Physical Review Research•Feb 25, 2026•0 citations
Abstract We review the nonlinear σ‐model approach to the localization problem. However, different from usual, we do not make use of replicas or superfields in order to perform the averaging over the realizations of the disorder. Instead, we pursue the derivation of the nonlinear σ‐model as an effective theory for the disordered electron system within the framework of the so‐called time‐path formalism, which is a real‐time finite‐temperature quantum field theory. This formalism is especially suitable for the description of the disordered electron system due to the automatic normalization of real‐time quantities, thereby avoiding the denominator problem. We shall compare this dynamic method with the replica‐ and superfield‐methods as the perturbative level of the calculation of physical quantities.
The interacting disordered electron problem is reviewed with emphasis on the quantum phase transitions that occur in a model system and on the field-theoretic methods used to describe them. An elementary discussion of conservation laws and diffusive dynamics is followed by a detailed derivation of the extended nonlinear sigma model, which serves as an effective field theory for the problem. A general scaling theory of metal-insulator and related transitions is developed, and explicit renormalization-group calculations for the various universality classes are reviewed and compared with experimental results. A discussion of pertinent physical ideas and phenomenological approaches to the metal-insulator transition not contained in the sigma-model approach is given, and phase-transition aspects of related problems, like disordered superconductors and the quantum Hall effect, are discussed. The review concludes with a list of open problems.
The effect of an electron heat bath on the motion of an electron system is studied using a quantum Langevin equation and deriving the influence functional of the Feynman-Vernon approach. The generating functional for the time-dependent, finite-temperature correlation functions of a disordered electron system with interactions is written down, and some of its symmetry properties are analyzed. Since the disorder only affects the time evolution of the system our formalism avoids the replica trick and provides a natural physical framework for studying a large variety of open problems in the field of disordered systems. As an example, we give an explicit expression for the averaged generating functional for a disordered electron system in the presence of a Coulomb interaction in terms of two complex matrix fields.
Abstract On the basis of the attractive Hubbard model with a Gaussian random potential we describe superconductivity in the presence of quenched disorder. A dynamic path‐integral formalism with an 8‐component Keldysh‐Nambu spinor is shown to be convenient for solving the averaging problem without the recourse to the replica trick. The effective action is bosonized in terms of collective variables, namely local Q‐matrices and real space pair fields. Saddle point values are calculated in two simplified limits.
We develop a dynamical approach based on the Schwinger-Keldysh formalism to derive a field-theoretic description of disordered and interacting electron systems. Within this formalism we calculate the perturbative renormalization-group (RG) equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein (Zh. \'Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]) using replicas. The major simplifying feature of this approach, as compared to Finkelstein's, is that instead of $\stackrel{\ensuremath{\rightarrow}}{N}0$ replicas, we only need to consider N=2N=2N=2 species. We compare the dynamical Schwinger-Keldysh approach and the replica methods, and we present a simple and pedagogical RG procedure to obtain Finkelstein's RG equations.
We introduce a quasiclassical Green's function approach describing the unitary yet irreversible dynamics of quantum systems effectively acting as their own environment. Combining a variety of concepts of quantum many-body theory, notably the nonlinear σ model of disordered systems, the G Σ formalism for strong correlations, and real-time path integration, the theory is capable of describing a wide range of system classes and disorder models. It extends previous work beyond perturbation theory (in inverse Hilbert-space dimensions), enabling a description of thermalization dynamics from short scattering times, through the onset of ergodicity at an effective “Thouless time,” up to the many-body Heisenberg time. We illustrate the approach with two case studies: (1) a brickwork model of unitarily coupled quantum circuits with and without conserved symmetries and (2) an array of capacitively coupled quantum dots. Using the spectral form factor as a test observable, we find good agreement with numerical simulations. We present our formalism in a self-contained and pedagogical manner, aiming to provide a transferable toolbox for the first-principles description of many-body chaotic quantum systems in regimes of strong entanglement.