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arXiv:1412.7942 · Physical Review B

Equilibration in a chiral Luttinger liquid

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

Published May 7, 2015• 12 citations
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Abstract

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 ${k}^{7}{T}^{7}$.

DOI
10.1103/physrevb.91.195110

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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.
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.
This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.
We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.
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.
Making use of refermionization techniques, we map the nonlinear chiral Luttinger liquid model of the edge modes of a spatially confined fractional quantum Hall cloud developed in our recent work [Nardin et al., Phys. Rev. A 107, 033320 (2023)] onto a one-dimensional system of massive and interacting chiral fermions, whose mass and interactions are set by the filling factor of the quantum Hall fluid and the shape of the external confining potential at the position of the edge. As an example of the predictive power of the refermionized theory, we report a detailed study of the dynamic structure factor and of the spectral function of a fractional quantum Hall cloud. Among other features, our refermionized theory provides a physical understanding of the effective decay of the edge excitations and of the universal power-law exponents at the thresholds of the dynamic structure factor. The quantitative accuracy of the refermionized theory is validated against a full two-dimensional calculation based on a combination of exact diagonalization and Monte Carlo sampling.