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arXiv:cond-mat/0510557 · Physical Review B

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

Jan von Delft, Florian Marquardt, Robert A. L. Smith, Vinay Ambegaokar

Published Nov 30, 2007• 24 citations

Abstract

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.

DOI: 10.1103/physrevb.76.195332

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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.

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.

Before we start applying path integration to treat Cooper pairing and superfluidity, it is a good idea to quickly review the concepts behind path integration. There are many textbooks providing plentiful details, such as Feynman’s seminal text [1] and Kleinert’s comprehensive compendium [2], and other works listed in the bibliography [3–5]. We will assume that the reader is already familiar with the basics of path-integral theory, so if the following paragraphs are not merely reminders to you, it is probably better to first consult these textbooks.

We investigate the description of excitonic effects within time-dependent density-functional theory (TDDFT). The exchange-correlation kernel

{f}_{xc}

introduced in TDDFT allows a clear separation of quasiparticle and excitonic effects. Using a diagrammatic representation for

{f}_{xc}

, we express its excitonic part

{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}

in terms of the effective vertex function $\ensuremath{\Lambda}$. The latter fulfills an integral equation that thereby establishes the exact correspondence between TDDFT and the standard many-body approach based on the Bethe-Salpeter equation (BSE). The diagrammatic structure of the kernel in the equation for $\ensuremath{\Lambda}$ suggests the possibility of strong cancellation effects. Should the cancellation take place, already the first-order approximation to

{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}

is sufficient. A potential advantage of TDDFT over the many-body BSE method is thus dependent on the efficiency of the above-quoted cancellation. We explicitly verify this for an analytically solvable two-dimensional two-band model. The calculations confirm that the low-order

{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}

perfectly describes the bound exciton as well as the excitonic effects in the continuous spectrum in a wide range of the electron-hole coupling strength.

Linear density response functions are calculated for symmetric nuclear matter of normal density by time-evolving two-time Green's functions in real time. Of particular interest is the effect of correlations. The system is therefore initially time-evolved with a collision term calculated in a direct Born approximation until fully correlated. An external time-dependent potential is then applied. The ensuing density fluctuations are recorded to calculate the density response. This method was previously used by Kwong and Bonitz for studying plasma oscillations in a correlated electron gas. The energy-weighted sum-rule for the response function is guaranteed by using conserving self-energy insertions as the method then generates the full vertex-functions. These can alternatively be calculated by solving a Bethe -Salpeter equation as done in works by Bozek et al. The (first order) mean field is derived from a momentum-dependent (non-local) interaction while 2 nd order self-energies are calculated using a particle-hole two-body effective (or residual) interaction given by a gaussian local potential. We show results of calculations of the response function S(ɷ, q 0 ) for q 0 = 0.2, 0.4 and 0.8 fm -1 . Comparison is made with the nucleons being un-correlated i.e. with only the first order mean field included. We discuss the relation of our work with the Landau quasi-particle theory as applied to nuclear systems by Babu and Brown and followers.