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arXiv:1107.4116 · Physical Review B

Equilibration of a spinless Luttinger liquid

К. А. Матвеев, Anton V. Andreev

Published Jan 19, 2012• 31 citations
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Abstract

We study how a Luttinger liquid of spinless particles in one dimension approaches thermal equilibrium. Full equilibration requires processes of backscattering of excitations, which occur at energies of the order of the bandwidth. Such processes are not accounted for by the Luttinger-liquid theory. We treat the high-energy excitations as mobile impurities and derive an expression for the equilibration rate in terms of their spectrum. Our results apply at any interaction strength.

DOI
10.1103/physrevb.85.041102

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We study how a Luttinger liquid of spinless particles in one dimension approaches thermal equilibrium. Full equilibration requires processes of backscattering of excitations, which occur at energies of the order of the bandwidth. Such processes are not accounted for by the Luttinger-liquid theory. We treat the high-energy excitations as mobile impurities and derive an expression for the equilibration rate in terms of their spectrum. Our results apply at any interaction strength.
We review some of the recent results on equilibration of one-dimensional quantum liquids. The low-energy properties of these systems are described by the Luttinger liquid theory, in which the excitations are bosonic quasiparticles. At low temperatures, the relaxation of the gas of excitations toward full equilibrium is exponentially slow. In electronic Luttinger liquids, these relaxation processes involve backscattering of electrons and give rise to interesting corrections to the transport properties of one-dimensional conductors. We focus on the phenomenological theory of the equilibration of a quantum liquid and obtain an expression for the relaxation rate in terms of the excitation spectrum.
Luttinger liquid theory accounts for the low-energy boson excitations of one-dimensional quantum liquids, but disregards the high-energy excitations. The most important high-energy excitations are holes, which have infinite lifetime at zero temperature. At finite temperatures, they can be scattered by thermally excited bosons. We describe the interaction of a hole with the bosons by treating it as a mobile impurity in a Luttinger liquid. This approach enables us to evaluate the scattering probability at arbitrary interaction strength. In general, the result is expressed in terms of the hole spectrum, its dependence on the density and momentum of the fluid, and the parameters of the Luttinger liquid Hamiltonian. In the special case of Galilean invariant systems, the scattering probability is expressed in terms of only the hole spectrum and its dependence on the fluid density. We apply our results to the problem of equilibration of one-dimensional quantum liquids.
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.
Luttinger liquid theory describes one-dimensional electron systems in terms of noninteracting bosonic excitations. In this approximation thermal excitations are decoupled from the current flowing through a quantum wire, and the conductance is quantized. We show that relaxation processes not captured by the Luttinger liquid theory lead to equilibration of the excitations with the current and give rise to a temperature-dependent correction to the conductance. In long wires, the magnitude of the correction is expressed in terms of the velocities of bosonic excitations. In shorter wires it is controlled by the relaxation rate.
We consider strongly interacting one-dimensional electron liquids where elementary excitations carry either spin or charge. At small temperatures a spinon created at the bottom of its band scatters off low-energy spin and charge excitations and follows the diffusive motion of a Brownian particle in momentum space. We calculate the mobility characterizing these processes and show that the resulting diffusion coefficient of the spinon is parametrically enhanced at low temperatures compared to that of a mobile impurity in a spinless Luttinger liquid. We briefly discuss that this hints at the relevance of spin in the process of equilibration of strongly interacting one-dimensional electrons, and comment on implications for transport in clean single-channel quantum wires.
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.