Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1404.0178 · Physical Review B

Relaxation in Luttinger liquids: Bose-Fermi duality

Ivan V. Protopopov, Dmitri B. Gutman, A. D. Mirlin

Published Sep 8, 2014• 29 citations
arXivHTMLPDF

Abstract

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 $1/{m}^{*}$ and $l$ characterize the curvature of the fermionic and bosonic spectra, respectively, and $T$ is the temperature.

DOI
10.1103/physrevb.90.125113

Related papers

Equilibration in a chiral Luttinger liquid

Ivan V. Protopopov, Dmitri B. Gutman, A. D. Mirlin

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 $T$ or weakly interacting bosons at high $T$. We calculate the equilibration rate which is found to scale with temperature as ${T}^{5}$ and ${T}^{14}$ in the high-temperature (``bosonic'') and the low-temperature (``fermonic'') regimes, respectively. The relaxation rate of a hot particle with the momentum $k$ in the fermonic regime scales as ${k}^{7}{T}^{7}$.

arXiv:1412.7942•Physical Review B•May 7, 2015•12 citations

KINETIC PROCESSES IN FERMI-LUTTINGER LIQUIDS

A. Levchenko, T. MICKLITZ

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.

Журнал Экспериментальной и Теоретической Физики•Jan 1, 2021•1 citation

Lifetime of excitations in a clean Luttinger liquid

Kirill V. Samokhin

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.

Journal of Physics Condensed Matter•Aug 10, 1998•71 citations

Kinetic Processes in Fermi–Luttinger Liquids

A. Levchenko, T. Micklitz

Abstract In this work we discuss extensions of the pioneering analysis by Dzyaloshinskii 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.

arXiv:2101.08737•Journal of Experimental and Theoretical Physics•Apr 1, 2021•3 citations

Dynamics of a heavy particle in a Luttinger liquid

A. H. Castro Neto, Matthew P. A. Fisher

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 ${\mathit{T}}^{\mathrm{*}}$=(m/M)${\mathit{E}}_{\mathit{F}}$, 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 ${\mathit{E}}_{\mathit{F}}$ the Fermi energy. Below ${\mathit{T}}^{\mathrm{*}}$, 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.

arXiv:cond-mat/9510094•Physical review. B, Condensed matter•Apr 15, 1996•90 citations

Relaxation processes in a disordered Luttinger liquid

Д. А. Багрец, I. V. Gornyi, A. D. Mirlin +1 more

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.

arXiv:0804.4887•Semiconductors•Aug 1, 2008•16 citations
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 \ensuremathλ=m∗l2T\ensuremath{\lambda}={m}^{*}{l}^{2}T\ensuremathλ=m∗l2T and the bosonic theory by \ensuremathλ\ensuremath−1{\ensuremath{\lambda}}^{\ensuremath{-}1}\ensuremathλ\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.
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.
Abstract In this work we discuss extensions of the pioneering analysis by Dzyaloshinskii 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.
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.
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.