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arXiv:cond-mat/9403043 · Physical review. B, Condensed matter

Mobility and diffusion of a particle coupled to a Luttinger liquid

A. H. Castro Neto, A. O. Caldeira

Published Aug 15, 1994• 21 citations
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Abstract

We study the mobility of a particle coupled to a one-dimensional interacting fermionic system, namely, a Luttinger liquid. We bosonize the Luttinger liquid and find the effective interaction between the particle and the bosonic system. We show that the dynamics of this system is completely equivalent to the acoustic polaron problem where the interaction has purely electronic origin. This problem has a zero-mode excitation, or soliton, in the strong-coupling limit, which corresponds to the the formation of a polarization cloud due to the fermion-fermion interaction around the particle. We find that, due to the scattering of the residual bosonic modes, the soliton has a finite mobility and diffusion coefficient at finite temperatures that depend on the fermion-fermion interaction. We show that at low temperatures the mobility and the diffusion coefficient are proportional to ${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ and ${\mathit{T}}^{5}$, respectively, and at high temperatures the mobility vanishes as ${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ while the diffusion increases as T.

DOI
10.1103/physrevb.50.4863

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We study the mobility of a particle coupled to a one-dimensional interacting fermionic system, namely, a Luttinger liquid. We bosonize the Luttinger liquid and find the effective interaction between the particle and the bosonic system. We show that the dynamics of this system is completely equivalent to the acoustic polaron problem where the interaction has purely electronic origin. This problem has a zero-mode excitation, or soliton, in the strong-coupling limit, which corresponds to the the formation of a polarization cloud due to the fermion-fermion interaction around the particle. We find that, due to the scattering of the residual bosonic modes, the soliton has a finite mobility and diffusion coefficient at finite temperatures that depend on the fermion-fermion interaction. We show that at low temperatures the mobility and the diffusion coefficient are proportional to T\ensuremath−4{\mathit{T}}^{\mathrm{\ensuremath{-}}4}T\ensuremath−4 and T5{\mathit{T}}^{5}T5, respectively, and at high temperatures the mobility vanishes as T\ensuremath−1{\mathit{T}}^{\mathrm{\ensuremath{-}}1}T\ensuremath−1 while the diffusion increases as T.
We study the dynamics of a heavy particle of mass M moving in a one-dimensional repulsively interacting Fermi gas. The Fermi gas is described using the Luttinger model and bosonization. By transforming to a frame comoving with the heavy particle, we map the model onto a generalized ``quantum impurity problem.'' A renormalization-group calculation reveals a crossover from strong to weak coupling upon scaling down in temperature. Above the crossover temperature scale T∗{\mathit{T}}^{\mathrm{*}}T∗=(m/M)EF{\mathit{E}}_{\mathit{F}}EF​, the particle's mobility \ensuremath{\mu} is found to be (roughly) temperature independent and proportional to the dimensionless conductance g, characterizing the one-dimensional Luttinger liquid. Here m(\ensuremath{\ll}M) is the fermion mass, and EF{\mathit{E}}_{\mathit{F}}EF​ the Fermi energy. Below T∗{\mathit{T}}^{\mathrm{*}}T∗, in the weak-coupling regime, the mobility grows and diverges as \ensuremath{\mu}(T)\ensuremath{\sim}${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ in the T\ensuremath{\rightarrow}0 limit. \textcopyright{} 1996 The American Physical Society.
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.
We study electronic transport in a Luttinger liquid with an embedded impurity, which is either a weak scatterer (WS) or a weak link (WL), when interacting electrons are coupled to one-dimensional massless bosons (e.g., acoustic phonons). We find that the duality relation, ΔWSΔWL=1, between scaling dimensions of the electron backscattering in the WS and WL limits, established for the standard Luttinger liquid, holds in the presence of the additional coupling for an arbitrary fixed strength of boson scattering from the impurity. This means that at low temperatures such a system remains either an ideal insulator or an ideal metal, regardless of the scattering strength. On the other hand, when fermion and boson scattering from the impurity are correlated, the system has a rich phase diagram that includes a metal-insulator transition at some intermediate values of the scattering.
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 explore the weak-strong-coupling Bose-Fermi duality in a model of a single-channel integer or fractional quantum Hall edge state with a finite-range interaction. The system is described by a chiral Luttinger liquid with nonlinear dispersion of bosonic and fermonic excitations. We use the bosonization, a unitary transformation, and a refermionization to map the system onto that of weakly interacting fermions at low temperature TTT or weakly interacting bosons at high TTT. We calculate the equilibration rate which is found to scale with temperature as T5{T}^{5}T5 and T14{T}^{14}T14 in the high-temperature (``bosonic'') and the low-temperature (``fermonic'') regimes, respectively. The relaxation rate of a hot particle with the momentum kkk in the fermonic regime scales as k7T7{k}^{7}{T}^{7}k7T7.
In this work we discuss extensions of the pioneering analysis by Dzyaloshinski ǐ and Larkin [Sov.Phys.JETP 38, 202 (1974)] of correlation functions for one-dimensional Fermi systems, focusing on the effects of quasiparticle relaxation enabled by a nonlinear dispersion.Throughout the work we employ both, the weakly interacting Fermi gas picture and nonlinear Luttinger liquid theory to describe attenuation of excitations and explore the fermion-boson duality between both approaches.Special attention is devoted to the role of spin-exchange processes, effects of interaction screening, and integrability.Thermalization rates for electron-and hole-like quasiparticles, as well as the decay rate of collective plasmon excitations and the momentum space mobility of spin excitations are calculated for various temperature regimes.The phenomenon of spin-charge drag is considered and the corresponding momentum transfer rate is determined.We further discuss how momentum relaxation due to several competing mechanisms, viz.triple electron collisions, electron-phonon scattering, and long-range inhomogeneities affect transport properties, and highlight energy transfer facilitated by plasmons from the perspective of the inhomogeneous Luttinger liquid model.Finally, we derive the full matrix of thermoelectric coefficients at the quantum critical point of the first conductance plateau transition, and address magnetoconductance in ballistic semiconductor nanowires with strong Rashba spin-orbit coupling.