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arXiv:0804.4294 · Physical Review Letters

Nonequilibrium Luttinger Liquid: Zero-Bias Anomaly and Dephasing

Dmitri B. Gutman, Yuval Gefen, A. D. Mirlin

Published Sep 16, 2008• 62 citations
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Abstract

A one-dimensional system of interacting electrons out of equilibrium is studied in the framework of the Luttinger liquid model. We analyze several setups and develop a theory of tunneling into such systems. A remarkable property of the problem is the absence of relaxation in energy distribution functions of left and right movers yet the presence of the finite dephasing rate due to electron-electron scattering, which smears zero-bias-anomaly singularities in the tunneling density of states.

DOI
10.1103/physrevlett.101.126802

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A one-dimensional system of interacting electrons out of equilibrium is studied in the framework of the Luttinger liquid model. We analyze several setups and develop a theory of tunneling into such systems. A remarkable property of the problem is the absence of relaxation in energy distribution functions of left and right movers yet the presence of the finite dephasing rate due to electron-electron scattering, which smears zero-bias-anomaly singularities in the tunneling density of states.
The Luttinger liquid model, which describes interacting electrons in a single-channel quantum wire, is completely integrable in the absence of disorder and as such does not exhibit any relaxation to equilibrium. We consider relaxation processes induced by inelastic electron-electron interactions in a disordered Luttinger liquid, focusing on the equilibration rate and its essential differences from the electron-electron scattering rate as well as the rate of phase relaxation. In the first part of the paper, we review the basic concepts in a disordered Luttinger liquid at equilibrium. These include the elastic renormalization, dephasing, and interference-induced localization. In the second part, we formulate a conceptually important framework for systematically studying the nonequilibrium properties of a strongly correlated (non-Fermi) Luttinger liquid. We derive a coupled set of kinetic equations for the fermionic and bosonic distribution functions that describe the evolution of a nonequilibrium Luttinger liquid. Remarkably, the energy equilibration rate in the conducting disordered quantum wire (at sufficiently high temperature, when the localization effects are suppressed by dephasing) is shown to be on the order of the rate of elastic scattering off disorder, independent of the interaction constant and temperature.
We develop a kinetic theory for strongly correlated disordered one-dimensional electron systems out of equilibrium within the Luttinger liquid model. In the absence of inhomogeneities, the model exhibits no relaxation to equilibrium. We derive kinetic equations for electron and plasmon distribution functions in the presence of impurities and calculate the equilibration rate ${\ensuremath{\gamma}}_{E}$. Remarkably, for not too low temperature and bias voltage, ${\ensuremath{\gamma}}_{E}$ is given by the elastic backscattering rate $\ensuremath{\gamma}$: the ratio ${\ensuremath{\gamma}}_{E}/\ensuremath{\gamma}\ensuremath{\sim}1$ is independent of the strength of electron-electron interaction, temperature, and bias.
Tunneling density of states (DOS) in Luttinger liquid has a dip at zero energy, commonly known as the zero-bias anomaly. In the presence of a magnetic field, in addition to the zero-bias anomaly, the DOS develops two peaks separated from the origin by the Zeeman energy. We show that these finite-bias anomalies are characterized by a power-law behavior of the DOS and the differential conductance, and find the corresponding exponents at arbitrary strength of the electron-electron interaction. The developed theory is applicable to various kinds of quantum wires, including carbon nanotubes.
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 $\ensuremath{\nu}(\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0,T)\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\sqrt{{\ensuremath{\epsilon}}^{*}/T})$.
We develop a theory of tunneling spectroscopy of interacting electrons in a nonequilibrium quantum wire coupled to reservoirs. The problem is modeled as an out-of-equilibrium Luttinger liquid with spatially dependent interaction. The interaction leads to the renormalization of the tunneling density of states, as well as to the redistribution of electrons over energies. Energy relaxation is controlled by plasmon scattering at the boundaries between regions with different interaction strengths and affects the distribution function of electrons in the wire as well as that of electrons emitted from the interacting regions into noninteracting electrodes.