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:2403.15930

Finite-frequency conductivity of nonlinear Luttinger liquid in smooth random potential

A. A. Dontsov, D. N. Aristov, А. П. Дмитриев

Published Mar 23, 2024• 0 citations
arXivHTMLPDF

Abstract

We analyze the uniform conductivity of a one-dimensional degenerate fermion system placed in a random disorder potential so smooth that backward scattering can be neglected. We use the nonlinear Luttinger liquid model to consider effects of both interaction and the curvature of fermionic dispersion. The finite frequency conductivity, calculated in the lowest order of disorder potential, consists of two parts. First one is the elastic contribution, largely independent of temperature and interaction. Second one is the inelastic contribution, strongly dependent on temperature and frequency and appearing upon simultaneous presence of curvature, disorder and interaction. We argue that apart from such finite frequency conductivity, there should always remain the $δ$-function peak of conductivity at zero frequency, whose weight is weakly dependent on the disorder.

DOI
10.48550/arxiv.2403.15930

Related papers

Umklapp process and resistivity in one-dimensional fermion systems

Thierry Giamarchi

The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.

Physical review. B, Condensed matter•Aug 15, 1991•222 citations

Effect of the long-range random potential on optical conductivity in a one-dimensional extended-Fermi-surface system

Sujit Sarkar

We study the effect of the long-range random potential on optical conductivity in a one-dimensional interacting electron system with extended-Fermi surface. We apply Abelian bosonization to show that in one part of the momentum space the system has a metallic behavior with a gap in the spin sector, while in the other region of momentum space it behaves like a gapless Luttinger liquid (paramagnetic metallic phase). Consequently we consider the scattering of electrons with the long-range random potential (forward scattering disorder) in that region of momentum space, where the system is in gapless Luttinger liquid phase. The real part of the optical conductivity has no Drude peak for small frequency $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}0),$ but the Drude weight is nonzero and it has the doping dependence, while the imaginary part of the optical conductivity shows the usual $1/\ensuremath{\omega}$ behavior, like in the clean system, and it has no doping dependence. We explain the doping dependence with the help of one-dimensional exact density of states.

Physical review. B, Condensed matter•Jan 1, 2001•0 citations

Dephasing and Weak Localization in Disordered Luttinger Liquid

I. V. Gornyi, A. D. Mirlin, Dmitry G. Polyakov

We study the transport properties of interacting electrons in a disordered quantum wire within the framework of the Luttinger liquid model. The conductivity at finite temperature is nonzero only because of inelastic electron-electron scattering. We demonstrate that the notion of weak localization is applicable to the strongly correlated one-dimensional electron system. We calculate the relevant dephasing rate, which for spinless electrons is governed by the interplay of electron-electron interaction and disorder, thus vanishing in the clean limit.

arXiv:cond-mat/0407305•Physical Review Letters•Jul 22, 2005•73 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

Frequency-temperature crossover in the conductivity of disordered Luttinger liquids

Bernd Rosenow, Andreas Glatz, Thomas Nattermann

The temperature $(T)$ and frequency $(\ensuremath{\omega})$ dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in $\ensuremath{\omega}∕T$ scaling of the conductivity, such scaling is violated in the RG treatment. We also determine the nonlinear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.

arXiv:cond-mat/0610426•Physical Review B•Oct 10, 2007•4 citations

Transport and elementary excitations of a Luttinger liquid

Gianaurelio Cuniberti, Maura Sassetti, Bernhard Krämer

The low-temperature AC conductance of a one-dimensional electron system with a strong interaction of finite range is calculated by using linear response theory. The conductance factorizes into parts which depend on the internal properties of the system, and the external probe. For short-range interaction, the result resembles that for non-interacting electrons, but with the zero-frequency limit and the Fermi velocity renormalized by the interaction strength. For strong and long-range interaction, the conductance shows a peak that is related to charge-wave excitations. In this limit, the AC conductance can be simulated by a quantum capacitance and a quantum inductance .

Journal of Physics Condensed Matter•Jan 8, 1996•28 citations
We analyze the uniform conductivity of a one-dimensional degenerate fermion system placed in a random disorder potential so smooth that backward scattering can be neglected. We use the nonlinear Luttinger liquid model to consider effects of both interaction and the curvature of fermionic dispersion. The finite frequency conductivity, calculated in the lowest order of disorder potential, consists of two parts. First one is the elastic contribution, largely independent of temperature and interaction. Second one is the inelastic contribution, strongly dependent on temperature and frequency and appearing upon simultaneous presence of curvature, disorder and interaction. We argue that apart from such finite frequency conductivity, there should always remain the δδδ-function peak of conductivity at zero frequency, whose weight is weakly dependent on the disorder.
The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.
We study the effect of the long-range random potential on optical conductivity in a one-dimensional interacting electron system with extended-Fermi surface. We apply Abelian bosonization to show that in one part of the momentum space the system has a metallic behavior with a gap in the spin sector, while in the other region of momentum space it behaves like a gapless Luttinger liquid (paramagnetic metallic phase). Consequently we consider the scattering of electrons with the long-range random potential (forward scattering disorder) in that region of momentum space, where the system is in gapless Luttinger liquid phase. The real part of the optical conductivity has no Drude peak for small frequency $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}0),$ but the Drude weight is nonzero and it has the doping dependence, while the imaginary part of the optical conductivity shows the usual $1/\ensuremath{\omega}$ behavior, like in the clean system, and it has no doping dependence. We explain the doping dependence with the help of one-dimensional exact density of states.
We study the transport properties of interacting electrons in a disordered quantum wire within the framework of the Luttinger liquid model. The conductivity at finite temperature is nonzero only because of inelastic electron-electron scattering. We demonstrate that the notion of weak localization is applicable to the strongly correlated one-dimensional electron system. We calculate the relevant dephasing rate, which for spinless electrons is governed by the interplay of electron-electron interaction and disorder, thus vanishing in the clean limit.
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.
The temperature (T)(T)(T) and frequency $(\ensuremath{\omega})$ dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in $\ensuremath{\omega}∕T$ scaling of the conductivity, such scaling is violated in the RG treatment. We also determine the nonlinear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.
The low-temperature AC conductance of a one-dimensional electron system with a strong interaction of finite range is calculated by using linear response theory. The conductance factorizes into parts which depend on the internal properties of the system, and the external probe. For short-range interaction, the result resembles that for non-interacting electrons, but with the zero-frequency limit and the Fermi velocity renormalized by the interaction strength. For strong and long-range interaction, the conductance shows a peak that is related to charge-wave excitations. In this limit, the AC conductance can be simulated by a quantum capacitance and a quantum inductance .