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Journal of Physics Conference Series

On a theory of low temperature electron decoherence in disordered conductors

Dmitry S. Golubev, Andrei D. Zaikin

Published Oct 1, 2008• 2 citations
Publisher

Abstract

Recently von Delft (JvD) attempted to revise our theory of low temperature decoherence of electrons in disordered conductors with electron-electron interactions [D.S. Golubev and A.D. Zaikin, Phys. Rev. B 59, 9195 (1999)] and argued [Intl. J. Mod. Phys. B 22, 727 (2008)] that by 'more accurate treatment of recoil efiects' he was able to obtain the electron decoherence rate that vanishes at T = 0. In this paper we demonstrate that this JvD's claim is in error, as it is based on ambiguous and uncontrolled manipulations violating basic principles of quantum theory, such as energy-time uncertainty relation, causality, fluctuation-dissipation theorem, detailed balance and the like. While our theory has a simple and transparent relation to the well known P ( E ) theory (which exactly accounts for all recoil efiects) JvD's approach is inconsistent with this theory thus being unable to correctly treat even recoil efiects it was explicitly aimed at.

DOI
10.1088/1742-6596/129/1/012016

Related papers

Comment on ``Influence Functional for Decoherence of Interacting Electrons in Disordered Conductors'' (cond-mat/0510563v1) and on related papers (cond-mat/0510556v1 and cond-mat/0510557v1)

Dmitry S. Golubev, Andrei D. Zaikin

Recently von Delft (cond-mat/0510563v1)(JvD) has successfully re-derived our influence functional for interacting electrons and claimed that within our approach he was able to obtain the electron decoherence rate that vanishes at T=0. In this Comment we demonstrate that this JvD's claim is in error, as it is based on ambiguous and uncontrolled manipulations violating basic principles of quantum theory, such as energy-time uncertainty relation, causality, fluctuation-dissipation theorem, detailed balance and the like. We also briefly address insufficient approximations employed by Marquardt {\it et al.} (cond-mat/0510556v1) and by von Delft {\it et al.} (cond-mat/0510557v1) and demonstrate that the results of all three papers in the limit T=0 are inconsistent with simple rules of algebra.

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In this review, we rederive the controversial influence functional approach of Golubev and Zaikin (GZ) for interacting electrons in disordered metals in a way that allows us to show its equivalence, before disorder averaging, to diagrammatic Keldysh perturbation theory. By representing a certain Pauli factor [Formula: see text] occuring in GZ's effective action in the frequency domain (instead of the time domain, as GZ do), we also achieve a more accurate treatment of recoil effects. With this change, GZ's approach reproduces, in a remarkably simple way, the standard, generally accepted result for the decoherence rate. — The main text and appendices A.1 to A.3 of the present review are comparatively brief, and have been published previously; for convenience, they are included here again (with minor revisions). The bulk of the review is contained in several additional, lengthy appendices containing the relevant technical details.

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We establish a connection between the influence functional approach of Golubev and Zaikin (GZ) and Keldysh diagrammatic perturbation theory for calculating the decoherence time τ ϕ of interacting electrons in disordered metals; we show how the standard diagrams for the Cooperon self energy can be recovered from GZ's influence functional e -(i SR + SI ) .This allows us to shed light on GZ's claim that SR is irrelevant for decoherence: SR generates as many important self energy diagrams as SI ; GZ's neglect of SR is permissible only at high temeratures (T /τ el ).

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Florian Marquardt, Jan von Delft, Robert A. L. Smith +1 more

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Quantum Decoherence and Weak Localization at Low Temperatures

Dmitry S. Golubev, Andrei D. Zaikin

We develope a theory of a fundamental effect of the interaction-induced decoherence of the electron wave function in a disordered metal. With the aid of the Keldysh technique and the path integral formalism we derive a formally exact equation of motion for the electron density matrix in the presence of interaction. We demonstrate that the effect of interaction of the electron with other electrons and lattice ions in a disordered metal is equivalent to that of an effective dissipative environment. Quantum noise of this environment causes quantum decoherence even at T=0. Our analysis explicitely accounts for the Pauli principle which plays an important role for inelastic scattering processes but turns out not to affect quantum decoherence. Our results seriously challenge the existence of strong localization in low dimensional disordered metals.

arXiv:cond-mat/9712203•Dec 17, 1997•0 citations
Recently von Delft (JvD) attempted to revise our theory of low temperature decoherence of electrons in disordered conductors with electron-electron interactions [D.S. Golubev and A.D. Zaikin, Phys. Rev. B 59, 9195 (1999)] and argued [Intl. J. Mod. Phys. B 22, 727 (2008)] that by 'more accurate treatment of recoil efiects' he was able to obtain the electron decoherence rate that vanishes at T = 0. In this paper we demonstrate that this JvD's claim is in error, as it is based on ambiguous and uncontrolled manipulations violating basic principles of quantum theory, such as energy-time uncertainty relation, causality, fluctuation-dissipation theorem, detailed balance and the like. While our theory has a simple and transparent relation to the well known P ( E ) theory (which exactly accounts for all recoil efiects) JvD's approach is inconsistent with this theory thus being unable to correctly treat even recoil efiects it was explicitly aimed at.
Recently von Delft (cond-mat/0510563v1)(JvD) has successfully re-derived our influence functional for interacting electrons and claimed that within our approach he was able to obtain the electron decoherence rate that vanishes at T=0. In this Comment we demonstrate that this JvD's claim is in error, as it is based on ambiguous and uncontrolled manipulations violating basic principles of quantum theory, such as energy-time uncertainty relation, causality, fluctuation-dissipation theorem, detailed balance and the like. We also briefly address insufficient approximations employed by Marquardt {\it et al.} (cond-mat/0510556v1) and by von Delft {\it et al.} (cond-mat/0510557v1) and demonstrate that the results of all three papers in the limit T=0 are inconsistent with simple rules of algebra.
In this review, we rederive the controversial influence functional approach of Golubev and Zaikin (GZ) for interacting electrons in disordered metals in a way that allows us to show its equivalence, before disorder averaging, to diagrammatic Keldysh perturbation theory. By representing a certain Pauli factor [Formula: see text] occuring in GZ's effective action in the frequency domain (instead of the time domain, as GZ do), we also achieve a more accurate treatment of recoil effects. With this change, GZ's approach reproduces, in a remarkably simple way, the standard, generally accepted result for the decoherence rate. — The main text and appendices A.1 to A.3 of the present review are comparatively brief, and have been published previously; for convenience, they are included here again (with minor revisions). The bulk of the review is contained in several additional, lengthy appendices containing the relevant technical details.
We point out that the low temperature saturation of the electron phase decoherence time in a disordered conductor can be explained within the existing theory of weak localization provided the effect of quantum (high frequency) fluctuations is taken into account. Making use of the fluctuation-dissipation theorem we evaluate the quantum decoherence time, the crossover temperature below which thermal effects become unimportant, and the weak localization correction $\ensuremath{\delta}\ensuremath{\sigma}$ at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0. For 1D systems the latter is found to be $\ensuremath{\delta}\ensuremath{\sigma}/\ensuremath{\sigma}\ensuremath{\propto}1/\sqrt{N}$, where NNN is the number of conducting channels.
We establish a connection between the influence functional approach of Golubev and Zaikin (GZ) and Keldysh diagrammatic perturbation theory for calculating the decoherence time τ ϕ of interacting electrons in disordered metals; we show how the standard diagrams for the Cooperon self energy can be recovered from GZ's influence functional e -(i SR + SI ) .This allows us to shed light on GZ's claim that SR is irrelevant for decoherence: SR generates as many important self energy diagrams as SI ; GZ's neglect of SR is permissible only at high temeratures (T /τ el ).
This is the first in a series of two papers, in which we revisit the problem of decoherence in weak localization. The basic challenge addressed in our work is to calculate the decoherence of electrons interacting with a quantum-mechanical environment while taking proper account of the Pauli principle. First, we review the usual influence functional approach valid for decoherence of electrons due to classical noise, showing along the way how the quantitative accuracy can be improved by properly averaging over closed (rather than unrestricted) random walks. We then use a heuristic approach to show how the Pauli principle may be incorporated into a path-integral description of decoherence in weak localization. This is accomplished by introducing an effective modification of the quantum noise spectrum, after which the calculation proceeds analogous to the case of classical noise. Using this simple but efficient method, which is consistent with much more laborious diagrammatic calculations, we demonstrate how the Pauli principle serves to suppress the decohering effects of quantum fluctuations of the environment, and essentially confirm the classic result of Altshuler, Aronov, and Khmelnitskii [J. Phys. C 15, 7367 (1982)] for the energy-averaged decoherence rate, which vanishes at zero temperature. Going beyond that, we employ our method to calculate explicitly the leading quantum corrections to the classical decoherence rates and to provide a detailed analysis of the energy dependence of the decoherence rate. The basic idea of our approach is general enough to be applicable to the decoherence of degenerate Fermi systems in contexts other than weak localization as well. Paper II will provide a more rigorous diagrammatic basis for our results by rederiving them from a Bethe-Salpeter equation for the Cooperon.
We develope a theory of a fundamental effect of the interaction-induced decoherence of the electron wave function in a disordered metal. With the aid of the Keldysh technique and the path integral formalism we derive a formally exact equation of motion for the electron density matrix in the presence of interaction. We demonstrate that the effect of interaction of the electron with other electrons and lattice ions in a disordered metal is equivalent to that of an effective dissipative environment. Quantum noise of this environment causes quantum decoherence even at T=0. Our analysis explicitely accounts for the Pauli principle which plays an important role for inelastic scattering processes but turns out not to affect quantum decoherence. Our results seriously challenge the existence of strong localization in low dimensional disordered metals.