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arXiv:cond-mat/0602411 · Physical Review B

From Luttinger liquid to Altshuler-Aronov anomaly in multichannel quantum wires

Christophe Mora, Reinhold Egger, Alexander Altland

Published Jan 9, 2007• 18 citations
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Abstract

A crossover theory connecting Altshuler-Aronov electron-electron interaction corrections and Luttinger liquid behavior in quasi-one-dimensional disordered conductors has been formulated. Based on an interacting nonlinear sigma model, we compute the tunneling density of states and the interaction correction to the conductivity, covering the full crossover.

DOI
10.1103/physrevb.75.035310

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A crossover theory connecting Altshuler-Aronov electron-electron interaction corrections and Luttinger liquid behavior in quasi-one-dimensional disordered conductors has been formulated. Based on an interacting nonlinear sigma model, we compute the tunneling density of states and the interaction correction to the conductivity, covering the full crossover.
Recent experiments have revealed that the temperature dependence of the conductance of quasiballistic quantum wires bears characteristic features of the Luttinger-liquid state. In this paper, the conductance of an N-channel quantum wire is calculated within the model of N coupled Luttinger liquids and under the assumption of weak disorder. It is shown that as the number of channels increases, a crossover from the Luttinger-liquid to the Fermi-liquid behavior occurs. This crossover manifests itself in the 1/N decrease of the scaling exponent of the temperature dependence. An exact expression for the scaling exponent for the case of N coupled Luttinger chains is obtained, and the large N limit is studied for the case of a quantum wire. The case of N=2 for electrons with spin is analyzed in detail, and a qualitative agreement with the experiment is achieved.
We calculate the low-energy tunneling density of states $\ensuremath{\nu}(\ensuremath{\epsilon},T)$ of an NNN-channel disordered wire, taking into account the electron-electron interaction nonperturbatively. The finite scattering rate $1/\ensuremath{\tau}$ results in a crossover from the Luttinger liquid behavior at higher energies, $\ensuremath{\nu}\ensuremath{\propto}{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}$, to the exponential dependence $\ensuremath{\nu}(\ensuremath{\epsilon},T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0)\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}{\ensuremath{\epsilon}}^{*}/\ensuremath{\epsilon})$ at low energies, where ${\ensuremath{\epsilon}}^{*}\ensuremath{\propto}1/(N\ensuremath{\tau})$. At finite temperature TTT, the tunneling density of states depends on the energy through the dimensionless variable $\ensuremath{\epsilon}/\sqrt{{\ensuremath{\epsilon}}^{*}T}$. At the Fermi level $\ensuremath{\nu}(\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0,T)\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\sqrt{{\ensuremath{\epsilon}}^{*}/T})$.
Properties of one-dimensional (1D) transport strongly depend on the proper choice of boundary conditions. It has been frequently stated that the Luttinger-liquid (LL) conductance is renormalized by the interaction as ge2{ge}^{2}ge2/h. To contest this result I develop a model of 1D LL wire with the interaction switching off at the infinities. Its solution shows that there is no renormalization of the universal conductance while the electrons have a free behavior in the source and drain reservoirs.
We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification g=2gcg=2{g}_{c}g=2gc​, where ggg is the interaction parameter in a spinless Luttinger liquid and gc{g}_{c}gc​ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.
We discuss the properties of interacting electrons on a finite chain with open boundary conditions. We extend the Haldane-Luttinger-liquid description to these systems and study how the presence of the boundaries modifies various correlation functions. In view of possible experimental applications to quantum wires, we analyze how tunneling measurements can reveal the underlying Luttinger-liquid properties. The two-terminal conductance is calculated. We also point out possible applications to quasi-one-dimensional materials and study the effects of magnetic impurities.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.