Interacting Electrons in Disordered Wires: Anderson Localization and Low-TTransport

Abstract

We study transport of interacting electrons in a low-dimensional disordered system at low temperature T . In view of localization by disorder, the conductivity σ(T ) may only be non-zero due to electron-electron scattering. For weak interactions, the weak-localization regime crosses over with lowering T into a dephasing-induced "power-law hopping". As T is further decreased, the Anderson localization in Fock space crucially affects σ(T ), inducing a transition at T = Tc, so that σ(T < Tc) = 0. The critical behavior of σ(T ) above Tc is ln σ(T ) ∝ −(T − Tc) −1/2 . The mechanism of transport in the critical regime is many-particle transitions between distant states in Fock space. 72.15.Rn, 71.30.+h, In a pathbreaking paper [1] Anderson demonstrated that a quantum particle may become localized by a random potential. In particular, in non-interacting systems of one-dimensional (1D) or two-dimensional (2D) geometry even weak disorder localizes all electronic states [2], thus leading to the exactly zero conductivity, σ(T ) = 0, whatever temperature T . A non-zero σ(T ) in such systems may only occur due to inelastic scattering processes leading to dephasing of electrons. Two qualitatively different sources of dephasing are possible: (i) scattering of electrons by external excitations (in practice, phonons) and (ii) electron-electron (e-e) scattering. In either case, at sufficiently high temperatures, the dephasing rate τ −1 φ is high, so that the localization effects are reduced to a weak-localization (WL) correction to the Drude conductivity. This correction behaves as ln τ φ in 2D and as τ 1/2 φ in quasi-1D (many-channel wire) systems [3], and thus diverges with lowering T , signaling the occurrence of the strong localization (SL) regime. This prompts a question as to how the system conducts at low T .For the case of electron-phonon scattering the answer is well known. The conductivity is then governed by Mott's variable-range hopping (VRH) [4], yielding σ(T ) ∝ exp{−(T 0 /T ) µ } with µ = 1/(d+1), where d is the spatial dimensionality. In the presence of a long-range Coulomb interaction, the Coulomb gap in the tunneling density of states modifies the VRH exponent, µ = 1 2 [5]. But what is the low-T behavior of σ(T ) if the electronphonon coupling is negligibly weak and the only source of the inelastic scattering is the e-e interaction? Our purpose here is to solve this long-standing fundamental problem, which is also of direct experimental relevance; see, e.g., Refs.[6] and [7,8], where the crossover from WL to SL with lowering T was studied for 1D and 2D systems, respectively. For definiteness, we concentrate on the case of a many-channel 1D system with a short-range interaction. Our results are, however, more general (including single-channel wires, 2D systems, Coulomb interaction), as we discuss in the end of the paper.It was proposed in [9] that the e-e interaction by itself is sufficient to induce VRH at low T . This idea was widely used for interpretation of experimental [8,10] and numerical [11] results on ...

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