Kinetic theory of fluctuations in conducting systems

Abstract

We propose an effective field theory describing the time dependent fluctuations of electrons in conducting systems, generalizing the well known kinetic theory of fluctuations. We apply then the theory to analyze the effects of strong electron-electron and electron-phonon scattering on the statistics of current fluctuations. We find that if the electron-electron scattering length is much shorter than the transport mean free path the higher cumulants of current are parametrically enhanced.Fluctuations of electric current in time have been the subject of fundamental and applied research since the time of Schottky. They have been a centerpiece in the study of systems ranging from vacuum tubes to quantum Hall bars (for a review see [1]). More recently noise has been recognized as an important diagnostic probe for the nature of disorder and interactions. Traditionally the number of electrons taking part in transport in macroscopic systems is large; for that reason current fluctuations obey nearly Gaussian statistics. Under such circumstances it is the power spectrum which is the object of primary interest when addressing the fluctuations. However, as the sample size shrinks down from a vacuum tube to a nanojunction, and the currents involved become correspondingly smaller, the central limit theorem does not apply any more, and information concerning higher current cumulants may be needed. Various techniques have been applied to address this issue [1,2,3,4,5,6].There is an interesting and relevant class of systems out of equilibrium in which the effect of quantum interference on transport quantities is negligible. Transport through such systems is often described by classical kinetic equations (e.g. the Boltzmann equation). The intensity of the noise spectrum can then found employing a related approach, known as the kinetic theory of fluctuations (KTF) [7,8,9]. The appeal of this approach is its intuitive transparency, yet traditionally it was constructed to describe pair correlation functions, hence was limited to finding the second current cumulant. In a recent insightful work Nagaev [10] has proposed a generalization of the KTF, expressing higher order cumulants in terms of pair correlators. Using this idea (which was termed the "cascade approach"), Nagaev calculated the third and the fourth cumulants of the current. While this approach was shown to successfully address a number of problems, its (quantum-mechanical) microscopic foundations, as well as its formulation in terms of an effective field theory (which would allow for further generalization, such as including interactions or calculating the full counting statistics (FCS)) were not fully understood. An important step in the latter direction was an introduction of a stochastic path integral by Pilgram, Jordan, Sukhorukov, and Büttiker [11].In the present paper we propose a phenomenological effective field theory describing the time dependent fluctuations in phase space of electrons in conducting systems. This is a field-theoretical implementation of the Nagae...

Nonequilibrium Zero-Bias Anomaly in Disordered Metals

“…Starting with the Keldysh non-linear σ-model, we derive an effective action that accounts for virtual fluctuations in disordered metals away from equilibrium. This action is complementary to the one for kinetics of real fluctuations (such as noise) developed earlier [11,12,13,14]. Further, we discuss a connection between our theory and phenomenological methods [7,8,9,10].…”

“…The theory [11,13,14] resulting from the σ-model (3), reduced to the subspace (5), accounts for real fluctuations in agreement with the cascade idea of Nagaev [17]. Below we use a similar ideology, "projecting" the σ-model on the subspace appropriate for the ZBA problem.…”

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