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arXiv:cond-mat/0210644 · Journal of the Physical Society of Japan

Decoherence of Interacting Electrons in Disordered Conductors: on the Relation between Influence Functional and Diagrammatic Approaches

Jan von Delft

Published Jan 3, 2003• 10 citations
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Abstract

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 ).

DOI
10.1143/jpsjs.72sa.24

Related papers

INFLUENCE FUNCTIONAL FOR DECOHERENCE OF INTERACTING ELECTRONS IN DISORDERED CONDUCTORS

JAN VON DELFT

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.

arXiv:cond-mat/0510563•International Journal of Modern Physics B•Mar 20, 2008•14 citations

On a theory of low temperature electron decoherence in disordered conductors

Dmitry S. Golubev, Andrei D. Zaikin

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.

Journal of Physics Conference Series•Oct 1, 2008•2 citations

Decoherence of interacting electrons in disordered conductors: on the relation between influence functional and diagrammatic approaches (Proceedings of the International Conference on Quantum Transport and Quantum Coherence(Localisation 2002))

Jan von Delft

Journal of the Physical Society of Japan•Jan 1, 2003•1 citation

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.

arXiv:cond-mat/0512411•Dec 16, 2005•1 citation

Decoherence in weak localization. II. Bethe-Salpeter calculation of the cooperon

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

This is the second in a series of two papers (Papers I and II) on the problem of decoherence in weak localization. In Paper I, we discussed how the Pauli principle could be incorporated into an influence functional approach for calculating the cooperon propagator and the magnetoconductivity. In the present paper, we check and confirm the results so obtained by diagrammatically setting up a Bethe-Salpeter equation for the cooperon, which includes self-energy and vertex terms on an equal footing and is free from both infrared and ultraviolet divergences. We then approximately solve this Bethe-Salpeter equation by the ansatz $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}(t)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}^{0}(t){e}^{\ensuremath{-}F(t)}$, where the decay function $F(t)$ determines the decoherence rate. We show that in order to obtain a divergence-free expression for the decay function $F(t)$, it is sufficient to calculate ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}^{1}(t)$, the cooperon in the position-time representation to first order in the interaction. Paper II is independent of Paper I and can be read without detailed knowledge of the latter.

arXiv:cond-mat/0510557•Physical Review B•Nov 30, 2007•24 citations

Decoherence in weak localization. I. Pauli principle in influence functional

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

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.

arXiv:cond-mat/0510556•Physical Review B•Nov 30, 2007•30 citations
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 ).
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.
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.
This is the second in a series of two papers (Papers I and II) on the problem of decoherence in weak localization. In Paper I, we discussed how the Pauli principle could be incorporated into an influence functional approach for calculating the cooperon propagator and the magnetoconductivity. In the present paper, we check and confirm the results so obtained by diagrammatically setting up a Bethe-Salpeter equation for the cooperon, which includes self-energy and vertex terms on an equal footing and is free from both infrared and ultraviolet divergences. We then approximately solve this Bethe-Salpeter equation by the ansatz $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}(t)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}^{0}(t){e}^{\ensuremath{-}F(t)}$, where the decay function F(t)F(t)F(t) determines the decoherence rate. We show that in order to obtain a divergence-free expression for the decay function F(t)F(t)F(t), it is sufficient to calculate ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}^{1}(t)$, the cooperon in the position-time representation to first order in the interaction. Paper II is independent of Paper I and can be read without detailed knowledge of the latter.
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.