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arXiv:cond-mat/0510563 · International Journal of Modern Physics B

INFLUENCE FUNCTIONAL FOR DECOHERENCE OF INTERACTING ELECTRONS IN DISORDERED CONDUCTORS

JAN VON DELFT

Published Mar 20, 2008• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217979208039083

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•Oct 24, 2008•0 citations
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 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 ).
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.
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.
xi The transport properties of a one dimensional disordered system of interacting spinpolarized electrons is studied using a bosonized field theory. In contrast to other approaches in that field [3], [2], a fully bosonized action on the Keldysh contour is used to calculate disorder averaged observables. After a brief introduction to Fermi and Luttinger liquids, we review bosonization, the Keldysh technique and disorder effects in mesoscopic systems. We calculate the first order correction to the clean conductivity in the disorder strength in order to develop a diagrammatic technique for a bosonized system on the Keldysh contour. Furthermore, the dissipative part of the self energy of the retarded plasmonic Green’s function is calculated via equations of motion in the high temperature regime. Corrections to the Drude conductivity of order = 1−K (weak interactions) and ωβ (high temperature) are calculated representing the terms beyond the saddle point approximation.
In this thesis we consider a variational wave function approach as a possible route to describe the competition between disorder and strong electron-electron interaction in two dimensions. In particular we aim to obtain a transparent and physically intuitive understanding of the competition between these two localizing forces within the simplest model where they both are active, namely the disordered Hubbard model at half filling and in a square lattice. Our approach is based on an approximate form of the ground-state wave function, which we believe contains the physically relevant ingredients for a correct description of both the Mott and the Anderson insulators, where electrons are localized by the Coulomb repulsion and by disorder, respectively. For strongly interacting fermionic systems, a standard variational wave function is constructed by a correlation term acting on a Slater determinant, the latter being an uncorrelated metallic state. Previous variational calculations showed that a long-range density-density correlation factor, so called Jastrow factor, is needed to correctly describe the Mott insulator [9]. This term, which is collective by definition, correlates spatially charge uctuations, thus preventing their free motion that would otherwise imply metallic conductance. For this reason, our variational wave function does include such a term. Anderson localization is instead mostly a matter of single-particle wave functions, hence it pertains to the uncorrelated Slater determinant which the Jastrow factor acts onto. We consider both the case in which we enforce paramagnetism in the wave function and the case in which we allow for magnetic ordering. Summarizing briefly our results, we find that, when the variational wave function is forced to be paramagnetic, the Anderson insulator to Mott insulator transition is continuous. This transition can be captured by studying several quantities. In particular, a novel one that we have identified and that is easily accessible variationally is the disconnected density-density fluctuation at long wavelength, defined by lim where ^nq is the Fourier transform of the charge density at momentum q, (...) denotes quantum average at fixed disorder and the overbar represents the average over disorder configurations. We find that Ndisc q!0 is everywhere finite in the Anderson insulator and vanishes critically at the Mott transition, staying zero in the Mott insulator. When magnetism is allowed and the hopping only connects nearest neighbor sites, upon increasing interaction the paramagnetic Anderson insulator first turns antiferromagnetic and finally the magnetic and compressible Anderson insulator gives way to an incompressible antiferromagnetic Mott insulator. The optimized uncorrelated Slater determinant is always found to be the eigenstate of a disordered non-interacting effective Hamiltonian, which suggests that the model is never metallic. Finally, when magnetism is frustrated by a next to nearest neighbor hopping, the overall sequence of phases does not change. However, the paramagnetic to magnetic transition within the Anderson insulator basin of stability turns first order. Indeed, within the magnetically ordered phase, we find many almost degenerate paramagnetic states with well defined local moments. This is suggestive of an emerging glassy behavior when the competition between disorder and strong correlation is maximum.