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

Physical Review A

Microscopic calculation of the nonlinear current fluctuations of a metallic resistor: The problem of heating in perturbation theory

A.–M. S. Tremblay, Bruce R. Patton, Paul C. Martin, Pierre F. Maldague

Published Apr 1, 1979• 52 citations
Publisher

Abstract

The effect of a constant electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}$ on the low-temperature Johnson noise of a metallic resistor is calculated by nonequilibrium diagrammatic perturbation theory. The expansion, in powers of ${E}^{2}$, is shown to diverge, and the divergence is associated with accumulating Joule heat. An insight into how to treat a steady state properly when there is heating is provided by the Boltzmann equation. A vertex function that corresponds to a steady-state solution of this equation is identified. With this vertex function it is possible to evaluate not only one-particle operators, for which the Boltzmann equation is sufficient, but also troublesome contributions to the current fluctuations and other two-body observables for which a many-body approach is required. The calculated corrections to the current correlations agree qualitatively, but not quantitatively, with simple phenomenological agruments. In particular, it is not possible to describe the leading effects of the electric field solely in terms of a variable local temperature. The model calculation illustrates and sheds some light on several problems that only appear when nonlinear deviations from thermal equilibrium must be considered: the complicated effects to which heat generation leads, the identification and treatment of these effects in perturbation theory, and the difficulties these effects pose for hypothetical generalizations of the fluctuation dissipation theorem. Some possible connections between the calculation and $\frac{1}{f}$ noise are also briefly explored. In particular, recent experiments that may confirm the conclusion of this calculation, that systems in which there is only impurity scattering do not exhibit $\frac{1}{f}$ noise, are noted.

DOI
10.1103/physreva.19.1721

Related papers

Dynamic perturbation theory

Uwe Claus Täuber

In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.

Cambridge University Press eBooks•Mar 6, 2014•1 citation

Divergences of the irreducible vertex functions in correlated metallic systems: Insights from the Anderson impurity model

Patrick Chalupa-Gantner, Patrik Gunacker, Thomas Schäfer +2 more

In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.

arXiv:1712.04171•Physical review. B./Physical review. B•Jun 22, 2018•80 citations

Quantum noise in current-biased Josephson junction

Yehoshua B. Levinson

We consider quantum fluctuations in a current biased overdamped Josephson junction in the regime when the bias current is larger than the critical current. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. Quantum effects are important when the resistor temperature is low compared to the Josephson frequency, even in the case when noise is measured at frequencies lower than the temperature. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and calculate the widths of the Josephson radiation lines.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 9, 2003•3 citations

Nonequilibrium dynamical mean-field theory of strongly correlated electrons

Volodymyr Turkowski, James K. Freericks

We present a review of our recent work in extending the successful dynamical mean-field theory from the equilibrium case to nonequilibrium cases. In particular, we focus on the problem of turning on a spatially uniform, but possibly time varying, electric field (neglecting all magnetic field effects). We show how to work with a manifestly gauge-invariant formalism, and compare numerical calculations from a transient-response formalism to different types of approximate treatments, including the semiclassical Boltzmann equation and perturbation theory in the interaction. In this review, we solve the nonequilibrium problem for the Falicov-Kimball model, which is the simplest many-body model and the easiest problem to illustrate the nonequilibrium behavior in both diffusive metals and Mott insulators. Due to space restrictions, we assume the reader already has some familiarity both with the Kadanoff-Baym-Keldysh nonequilibrium formalism and with equilibrium dynamical mean-field theory; we provide a guide to the literature where additional details can be found.

arXiv:cond-mat/0612466•WORLD SCIENTIFIC eBooks•Jul 1, 2007•12 citations

Fluctuations and correlations

Tero Tapio Heikkilä

This chapter deals with noise: time-dependent fluctuation of observables at or out of equilibrium. The main emphasis is on current noise. The chapter explains the use of the correlation function, the fluctuation-dissipation theorem, and thermal and vacuum fluctuations. It describes shot noise produced in out-of-equilibrium setups with a simple Schottky model, with the microscopic scattering and Boltzmann-Langevin approaches. The chapter also discusses the Langevin approach of describing coupling of noise in electrical circuits and a related description of noise measurements, and briefly explains the theory for cross correlations. The second part of the chapter discusses effects of noise on quantum systems – energy relaxation and dephasing. The chapter concludes by presenting the theory of full counting statistics and briefly describes heat current noise.

Oxford University Press eBooks•Jan 31, 2013•2 citations

Nonequilibrium conductivity at quantum critical points

A. M. Berridge, Andrew G. Green

Quantum criticality provides an important route to revealing universal nonequilibrium behavior. A canonical example of a critical point is the Bose-Hubbard model, which we study under the application of an electric field. A Boltzmann transport formalism and $\ensuremath{\epsilon}$ expansion are used to obtain the nonequilibrium conductivity and current noise. This approach allows us to explicitly identify how a universal nonequilibrium steady state is maintained, by identifying the rate-limiting step in balancing Joule heating and dissipation to a heat bath. It also reveals that the nonequilibrium distribution function is very far from a thermal distribution.

arXiv:1312.4432•Physical Review B•Dec 30, 2013•5 citations
The effect of a constant electric field E\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}E\ensuremath→​ on the low-temperature Johnson noise of a metallic resistor is calculated by nonequilibrium diagrammatic perturbation theory. The expansion, in powers of E2{E}^{2}E2, is shown to diverge, and the divergence is associated with accumulating Joule heat. An insight into how to treat a steady state properly when there is heating is provided by the Boltzmann equation. A vertex function that corresponds to a steady-state solution of this equation is identified. With this vertex function it is possible to evaluate not only one-particle operators, for which the Boltzmann equation is sufficient, but also troublesome contributions to the current fluctuations and other two-body observables for which a many-body approach is required. The calculated corrections to the current correlations agree qualitatively, but not quantitatively, with simple phenomenological agruments. In particular, it is not possible to describe the leading effects of the electric field solely in terms of a variable local temperature. The model calculation illustrates and sheds some light on several problems that only appear when nonlinear deviations from thermal equilibrium must be considered: the complicated effects to which heat generation leads, the identification and treatment of these effects in perturbation theory, and the difficulties these effects pose for hypothetical generalizations of the fluctuation dissipation theorem. Some possible connections between the calculation and 1f\frac{1}{f}f1​ noise are also briefly explored. In particular, recent experiments that may confirm the conclusion of this calculation, that systems in which there is only impurity scattering do not exhibit 1f\frac{1}{f}f1​ noise, are noted.
In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.
In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.
We consider quantum fluctuations in a current biased overdamped Josephson junction in the regime when the bias current is larger than the critical current. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. Quantum effects are important when the resistor temperature is low compared to the Josephson frequency, even in the case when noise is measured at frequencies lower than the temperature. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and calculate the widths of the Josephson radiation lines.
We present a review of our recent work in extending the successful dynamical mean-field theory from the equilibrium case to nonequilibrium cases. In particular, we focus on the problem of turning on a spatially uniform, but possibly time varying, electric field (neglecting all magnetic field effects). We show how to work with a manifestly gauge-invariant formalism, and compare numerical calculations from a transient-response formalism to different types of approximate treatments, including the semiclassical Boltzmann equation and perturbation theory in the interaction. In this review, we solve the nonequilibrium problem for the Falicov-Kimball model, which is the simplest many-body model and the easiest problem to illustrate the nonequilibrium behavior in both diffusive metals and Mott insulators. Due to space restrictions, we assume the reader already has some familiarity both with the Kadanoff-Baym-Keldysh nonequilibrium formalism and with equilibrium dynamical mean-field theory; we provide a guide to the literature where additional details can be found.
This chapter deals with noise: time-dependent fluctuation of observables at or out of equilibrium. The main emphasis is on current noise. The chapter explains the use of the correlation function, the fluctuation-dissipation theorem, and thermal and vacuum fluctuations. It describes shot noise produced in out-of-equilibrium setups with a simple Schottky model, with the microscopic scattering and Boltzmann-Langevin approaches. The chapter also discusses the Langevin approach of describing coupling of noise in electrical circuits and a related description of noise measurements, and briefly explains the theory for cross correlations. The second part of the chapter discusses effects of noise on quantum systems – energy relaxation and dephasing. The chapter concludes by presenting the theory of full counting statistics and briefly describes heat current noise.
Quantum criticality provides an important route to revealing universal nonequilibrium behavior. A canonical example of a critical point is the Bose-Hubbard model, which we study under the application of an electric field. A Boltzmann transport formalism and $\ensuremath{\epsilon}$ expansion are used to obtain the nonequilibrium conductivity and current noise. This approach allows us to explicitly identify how a universal nonequilibrium steady state is maintained, by identifying the rate-limiting step in balancing Joule heating and dissipation to a heat bath. It also reveals that the nonequilibrium distribution function is very far from a thermal distribution.