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Journal of Physics Condensed Matter

Lifetime of excitations in a clean Luttinger liquid

Kirill V. Samokhin

Published Aug 10, 1998• 71 citations
Publisher

Abstract

Non-linear band dispersion in the Luttinger model of one-dimensional interacting electrons gives rise to collisions between the bosonic sound-like excitations. The decay rate is calculated beyond a plain perturbation theory both at T = 0 and . A self-consistent approach reveals a non-analytical dependence of the lifetime of excitations on the coupling constant, wave vector and temperature.

DOI
10.1088/0953-8984/10/31/002

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Non-linear band dispersion in the Luttinger model of one-dimensional interacting electrons gives rise to collisions between the bosonic sound-like excitations. The decay rate is calculated beyond a plain perturbation theory both at T = 0 and . A self-consistent approach reveals a non-analytical dependence of the lifetime of excitations on the coupling constant, wave vector and temperature.
We consider spinless electrons in two dimensions with the bare spectrum ${\mathrm{\ensuremath{\epsilon}}}_{\mathbf{p}}$=vF{\mathit{v}}_{\mathit{F}}vF​(\ensuremath{\Vert}px{\mathit{p}}_{\mathit{x}}px​\ensuremath{\Vert}+\ensuremath{\Vert}py{\mathit{p}}_{\mathit{y}}py​\ensuremath{\Vert}). In momentum space, the interactions among electrons have a finite range q0{\mathit{q}}_{0}q0​, which is small compared to the Fermi momentum. A golden-rule calculation of the electron lifetime indictes a breakdown of Landau's Fermi-liquid theory in the model. At the one-loop level of perturbation theory, we show that the density wave and the superconducting instabilities cancel each other and there is no symmetry breaking. We solve the model via bosonization; the excitation spectrum is found to consist of gapless bosonic modes as in a one-dimensional Luttinger liquid.
We explore the lifetime of excitations in a dispersive Luttinger liquid. We perform a bosonization supplemented by a sequence of unitary transformations that allows us to treat the problem in terms of weakly interacting quasiparticles. The relaxation described by the resulting Hamiltonian is analyzed by bosonic and (after a refermionization) by fermionic perturbation theory. We show that the fermionic and bosonic formulations of the problem exhibit a remarkable strong-weak coupling duality. Specifically, the fermionic theory is characterized by a dimensionless coupling constant $\ensuremath{\lambda}={m}^{*}{l}^{2}T$ and the bosonic theory by ${\ensuremath{\lambda}}^{\ensuremath{-}1}$, where 1/m∗1/{m}^{*}1/m∗ and lll characterize the curvature of the fermionic and bosonic spectra, respectively, and TTT is the temperature.
For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. Recent progress in understanding 1D quantum fluids beyond the low-energy limit is reviewed, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi-edge singularity and the Fermi-liquid theory, perturbation theory in the interaction strength, and new ways of treating finite-size properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. Existing results for various dynamic correlation functions are reviewed, in particular, the dynamic structure factor and the spectral function. Moreover, it is shown how a dispersion nonlinearity leads to finite particle lifetimes and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and the relation between the two is explained.
The low-energy properties of one-dimensional quantum liquids are commonly described in terms of the Tomonaga-Luttinger liquid theory, in which the elementary excitations are free bosons. To this approximation, the theory can be alternatively recast in terms of free fermions. In both approaches, small perturbations give rise to finite lifetimes of excitations. We evaluate the decay rate of fermionic excitations and show that it scales as the eighth power of energy, in contrast to the much faster decay of bosonic excitations. Our results can be tested experimentally by measuring the broadening of power-law features in the density structure factor or spectral functions.
One-dimensional electrons with a linearized dispersion relation are equivalent to a collection of harmonic plasmon modes, which represent long wavelength density oscillations. An immediate consequence of this Luttinger model of one-dimensional electron systems is the absence of inelastic scattering processes responsible for the relaxation of nonequilibrium states. In a generic nonlinear Luttinger liquid, plasmons may decay and thus acquire a finite lifetime. We show that equilibration of plasmons is hierarchical and has profound implications for the dynamics after a thermal quench. We also develop a thermal transport theory and compute thermal conductance of the nonlinear Luttinger liquid by treating the collision integral of plasmons in a manifestly nonperturbative way.
We study inelastic decay of bosonic excitations in a Luttinger liquid. In a model with a linear excitation spectrum the decay rate diverges. We show that this difficulty is resolved when the interaction between constituent particles is strong, and the excitation spectrum is nonlinear. Although at low energies the nonlinearity is weak, it regularizes the divergence in the decay rate. We develop a theoretical description of the approach of the system to thermal equilibrium. The typical relaxation rate scales as the fifth power of temperature.