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:2605.21774 · Physical review. B./Physical review. B

Conductivity of a non-Galilean-invariant Fermi liquid: Exact solution of the kinetic equation

Tatia Kiliptari, Vladimir I. Yudson, Dmitrii L. Maslov

Published Oct 14, 2026• 0 citations
arXivHTMLPDF

Abstract

We obtain an exact expression for the conductivity of a disordered, non-Galilean-invariant Fermi liquid by solving the kinetic equation with both screened Coulomb and $z=3$ Pomeranchuk critical interactions. While consistent with previous asymptotic results, our solution shows that electron-electron interactions enter the conductivity solely via the quasiparticle scattering time, $τ_\mathrm{ee}$. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when $1/τ_\mathrm{ee}$ becomes comparable to the larger of the impurity scattering rate and the probe frequency, $Ω$. In addition, the exact solution yields the optical response in the hydrodynamic regime, $Ω\ll 1/τ_\mathrm{ee}$, which is inaccessible within perturbation theory. Near a $z=3$ Pomeranchuk quantum critical point, consistency between the kinetic-equation and Kubo approaches requires proper inclusion of mass renormalization within the Eliashberg approximation, which also ensures that the crossover between the collisionless and hydrodynamic regimes in the optical conductivity occurs at the Planckian scale $Ω\sim T$.

DOI
10.1103/d791-lz1h

Related papers

Hydrodynamic transport in the Luttinger-Abrikosov-Beneslavskii non-Fermi liquid

Julia M. Link, Igor F. Herbut

We determine the shear viscosity and the dc electrical conductivity of interacting three-dimensional Luttinger semimetals, which have a quadratic band touching point in the energy spectrum, in the hydrodynamic regime. It is well known that when the chemical potential is right at the band touching point, the long-range Coulomb interaction induces the Luttinger-Abrikosov-Beneslavskii (LAB) phase at $T=0$, which is an interacting, scale-invariant, non-Fermi-liquid state of electrons. Upon combining the renormalization-group (RG) analysis near the upper critical spatial dimension of 4 with the Boltzmann kinetic equation, we determine the universal ratio of viscosity over entropy and the electrical dc conductivity of the system at the interacting LAB fixed point of the RG flow. The projection of the Coulomb interaction on the eigenstates of the system is found to play an important quantitative role for the scattering amplitude in the collision integral, and the so-called Auger processes make a large numerical contribution to the inverse scattering time in the transport quantities. The obtained leading-order result suggests that the universal ratio of the viscosity over entropy, when extrapolated to the physical three spatial dimensions, is above, but could be rather close to, the Kovtun-Son-Starinets lower bound.

arXiv:1911.06431•Physical review. B./Physical review. B•Mar 26, 2020•12 citations

Conserving relaxation-time approximation for electron-electron collisions

Sayak Bhattacharjee, Tatia Kiliptari, Dmitrii L. Maslov

We develop a conserving relaxation-time approximation (cRTA) based on an explicit energy-resolved projection onto the full space of collision invariants. Our cRTA retains the energy dependence of the nonequilibrium quasiparticle distribution, allowing one to describe transport quantities sensitive to states near, but not exactly on, the Fermi surface (FS). We apply the method to several charge-transport problems in both Galilean-invariant and non-Galilean-invariant Fermi liquids. In particular, the cRTA reproduces the low- and high-temperature limits of the dc conductivity of a non-Galilean-invariant Fermi liquid with disorder, the hydrodynamic and collisionless limits of the finite-wavevector longitudinal conductivity of a clean Galilean-invariant Fermi liquid, and the asymptotic scaling forms of the optical conductivity of a clean non-Galilean-invariant Fermi liquid beyond the semiclassical limit. For several observables, the agreement with exact solutions is quantitative at the percent level. These results demonstrate that the cRTA provides a simple and accurate framework for describing transport beyond the FS projection.

arXiv:2605.26207•May 25, 2026•0 citations

Finite-temperature Fermi-liquid theory of electrical conductivity

Tadashi Toyoda

The Fermi-liquid transport theory of electrical conductivity formulated by Eliashberg [Zh. Eksp. Teor. Fiz. 41, 1241 (1961)] is further developed, using the analytic continuation of the finite-temperature Ward-Takahashi relations obtained by the present author [Ann. Phys. (N.Y.) 173, 226 (1987)] to maintain the theoretical consistency of approximation schemes for solving the Bethe-Salpeter equation. In particular, the finite-temperature Ward-Takahashi relations corresponding to the local conservation of the electron number density and to the conservation of the electron momentum are considered. A general expression for the electrical conductivity in terms of the proper vertex part is also obtained. In the formula, cancellation of the quasiparticle renormalization factors at the Fermi surface is explicitly shown in a general way without assuming a particular model. In the case that the electrons are scattered by phonons, the formula gives the Gr\"uneisen-Bloch formula for the on-shell phonons. For the impurity resistivity, the theory of Edwards [Philos. Mag. 3, 1020 (1958)] is developed to finite temperature by introducing a hypothetical boson field that replaces the impurity averaging.

Physical Review A•Mar 1, 1989•22 citations

Long-wavelength correlations and transport in a marginal Fermi liquid

Arkady Shekhter, Chandra Mohan Varma

Marginal Fermi liquid was originally introduced as a phenomenological description of the cuprates in a part of the metallic doping range which appears to be governed by fluctuations due to a quantum-critical point. An essential result due to the form of the assumed fluctuation spectra is that the large inelastic quasiparticle relaxation rate near the Fermi surface is proportional to the energy measured from the chemical potential, ${\ensuremath{\tau}}_{i}^{\ensuremath{-}1}\ensuremath{\propto}\ensuremath{\epsilon}$. We present a microscopic long-wavelength derivation of the hydrodynamic properties in such a situation by an extension of the procedure that Eliashberg used for the derivation of the hydrodynamic properties of a Landau-Fermi liquid. In particular, the density-density and the current-current correlations and the relation between the two are derived, and the connection to microscopic calculations of the frequency dependence of the optical conductivity with an additional Fermi-liquid correction factor shown to follow. The method used here may be necessary, quite generally, for the correct hydrodynamic theory for any problem of quantum-critical fluctuations in fermions.

arXiv:0809.1104•Physical Review B•Jan 23, 2009•20 citations

Transport in Disordered Many-Body Systems

Giancarlo Calvanese Strinati, Claudio Castellani, C. Di Castro

A description of quantum transport for electrons in the presence of a random impurity potential is provided in terms of a kinetic equation, which is obtained from the diagrammatic structure for the linear response functions. When the electron-electron interaction is neglected, the kinetic equation is of the Boltzmann type, with an effective scattering kernel that signals localization effects. With the further inclusion of the interaction, a suitable distribution function for the energy of the quasiparticles can be introduced even in the presence of strong disorder. In this case the kinetic equation has the diffusive form, with diffusion coefficient and Landau interaction parameters that get renormalized by the interplay of disorder and interaction. The quasiparticles kinetic equation is also shown to contain all known results for the hydrodynamic correlation functions.

Physica Scripta•Jan 1, 1989•0 citations

Dynamical susceptibility of a Fermi liquid

Vladimir A. Zyuzin, Prachi Sharma, Dmitrii L. Maslov

We study dynamic response of a Fermi liquid in the spin, charge, and nematic channels beyond the random phase approximation for the dynamically screened Coulomb potential. In all the channels, one-loop order corrections to the irreducible susceptibility result in a nonzero spectral weight of the corresponding fluctuations above the particle-hole continuum boundary. It is shown that the imaginary part of the spin susceptibility, $\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})$, falls off as ${q}^{2}/\ensuremath{\omega}$ for frequencies above the continuum boundary ($\ensuremath{\omega}\ensuremath{\gg}{v}_{\mathrm{F}}q$) and below the model-dependent cutoff frequency, whereas the imaginary part of the charge susceptibility, $\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})$, falls off as ${(q/{k}_{F})}^{2}{q}^{2}/\ensuremath{\omega}$ for frequencies above the plasma frequency. An extra factor of ${(q/{k}_{F})}^{2}$ in $\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})$ as compared to $\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})$ is a direct consequence of Galilean invariance. The imaginary part of the nematic susceptibility increases linearly with $\ensuremath{\omega}$ up to a peak at the ultraviolet energy scale---the plasma frequency and/or Fermi energy---and then decreases with $\ensuremath{\omega}$. We also obtain explicit forms of the spin susceptibility from the kinetic equation in the collisionless limit and for the Landau function that contains up to the first three harmonics.

arXiv:1806.10159•Physical review. B./Physical review. B•Sep 21, 2018•13 citations
We obtain an exact expression for the conductivity of a disordered, non-Galilean-invariant Fermi liquid by solving the kinetic equation with both screened Coulomb and z=3z=3z=3 Pomeranchuk critical interactions. While consistent with previous asymptotic results, our solution shows that electron-electron interactions enter the conductivity solely via the quasiparticle scattering time, τeeτ_\mathrm{ee}τee​. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when 1/τee1/τ_\mathrm{ee}1/τee​ becomes comparable to the larger of the impurity scattering rate and the probe frequency, ΩΩΩ. In addition, the exact solution yields the optical response in the hydrodynamic regime, Ω≪1/τeeΩ\ll 1/τ_\mathrm{ee}Ω≪1/τee​, which is inaccessible within perturbation theory. Near a z=3z=3z=3 Pomeranchuk quantum critical point, consistency between the kinetic-equation and Kubo approaches requires proper inclusion of mass renormalization within the Eliashberg approximation, which also ensures that the crossover between the collisionless and hydrodynamic regimes in the optical conductivity occurs at the Planckian scale Ω∼TΩ\sim TΩ∼T.
We determine the shear viscosity and the dc electrical conductivity of interacting three-dimensional Luttinger semimetals, which have a quadratic band touching point in the energy spectrum, in the hydrodynamic regime. It is well known that when the chemical potential is right at the band touching point, the long-range Coulomb interaction induces the Luttinger-Abrikosov-Beneslavskii (LAB) phase at T=0T=0T=0, which is an interacting, scale-invariant, non-Fermi-liquid state of electrons. Upon combining the renormalization-group (RG) analysis near the upper critical spatial dimension of 4 with the Boltzmann kinetic equation, we determine the universal ratio of viscosity over entropy and the electrical dc conductivity of the system at the interacting LAB fixed point of the RG flow. The projection of the Coulomb interaction on the eigenstates of the system is found to play an important quantitative role for the scattering amplitude in the collision integral, and the so-called Auger processes make a large numerical contribution to the inverse scattering time in the transport quantities. The obtained leading-order result suggests that the universal ratio of the viscosity over entropy, when extrapolated to the physical three spatial dimensions, is above, but could be rather close to, the Kovtun-Son-Starinets lower bound.
We develop a conserving relaxation-time approximation (cRTA) based on an explicit energy-resolved projection onto the full space of collision invariants. Our cRTA retains the energy dependence of the nonequilibrium quasiparticle distribution, allowing one to describe transport quantities sensitive to states near, but not exactly on, the Fermi surface (FS). We apply the method to several charge-transport problems in both Galilean-invariant and non-Galilean-invariant Fermi liquids. In particular, the cRTA reproduces the low- and high-temperature limits of the dc conductivity of a non-Galilean-invariant Fermi liquid with disorder, the hydrodynamic and collisionless limits of the finite-wavevector longitudinal conductivity of a clean Galilean-invariant Fermi liquid, and the asymptotic scaling forms of the optical conductivity of a clean non-Galilean-invariant Fermi liquid beyond the semiclassical limit. For several observables, the agreement with exact solutions is quantitative at the percent level. These results demonstrate that the cRTA provides a simple and accurate framework for describing transport beyond the FS projection.
The Fermi-liquid transport theory of electrical conductivity formulated by Eliashberg [Zh. Eksp. Teor. Fiz. 41, 1241 (1961)] is further developed, using the analytic continuation of the finite-temperature Ward-Takahashi relations obtained by the present author [Ann. Phys. (N.Y.) 173, 226 (1987)] to maintain the theoretical consistency of approximation schemes for solving the Bethe-Salpeter equation. In particular, the finite-temperature Ward-Takahashi relations corresponding to the local conservation of the electron number density and to the conservation of the electron momentum are considered. A general expression for the electrical conductivity in terms of the proper vertex part is also obtained. In the formula, cancellation of the quasiparticle renormalization factors at the Fermi surface is explicitly shown in a general way without assuming a particular model. In the case that the electrons are scattered by phonons, the formula gives the Gr\"uneisen-Bloch formula for the on-shell phonons. For the impurity resistivity, the theory of Edwards [Philos. Mag. 3, 1020 (1958)] is developed to finite temperature by introducing a hypothetical boson field that replaces the impurity averaging.
Marginal Fermi liquid was originally introduced as a phenomenological description of the cuprates in a part of the metallic doping range which appears to be governed by fluctuations due to a quantum-critical point. An essential result due to the form of the assumed fluctuation spectra is that the large inelastic quasiparticle relaxation rate near the Fermi surface is proportional to the energy measured from the chemical potential, ${\ensuremath{\tau}}_{i}^{\ensuremath{-}1}\ensuremath{\propto}\ensuremath{\epsilon}$. We present a microscopic long-wavelength derivation of the hydrodynamic properties in such a situation by an extension of the procedure that Eliashberg used for the derivation of the hydrodynamic properties of a Landau-Fermi liquid. In particular, the density-density and the current-current correlations and the relation between the two are derived, and the connection to microscopic calculations of the frequency dependence of the optical conductivity with an additional Fermi-liquid correction factor shown to follow. The method used here may be necessary, quite generally, for the correct hydrodynamic theory for any problem of quantum-critical fluctuations in fermions.
A description of quantum transport for electrons in the presence of a random impurity potential is provided in terms of a kinetic equation, which is obtained from the diagrammatic structure for the linear response functions. When the electron-electron interaction is neglected, the kinetic equation is of the Boltzmann type, with an effective scattering kernel that signals localization effects. With the further inclusion of the interaction, a suitable distribution function for the energy of the quasiparticles can be introduced even in the presence of strong disorder. In this case the kinetic equation has the diffusive form, with diffusion coefficient and Landau interaction parameters that get renormalized by the interplay of disorder and interaction. The quasiparticles kinetic equation is also shown to contain all known results for the hydrodynamic correlation functions.
We study dynamic response of a Fermi liquid in the spin, charge, and nematic channels beyond the random phase approximation for the dynamically screened Coulomb potential. In all the channels, one-loop order corrections to the irreducible susceptibility result in a nonzero spectral weight of the corresponding fluctuations above the particle-hole continuum boundary. It is shown that the imaginary part of the spin susceptibility, $\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})$, falls off as ${q}^{2}/\ensuremath{\omega}$ for frequencies above the continuum boundary ($\ensuremath{\omega}\ensuremath{\gg}{v}_{\mathrm{F}}q$) and below the model-dependent cutoff frequency, whereas the imaginary part of the charge susceptibility, $\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})$, falls off as ${(q/{k}_{F})}^{2}{q}^{2}/\ensuremath{\omega}$ for frequencies above the plasma frequency. An extra factor of (q/kF)2{(q/{k}_{F})}^{2}(q/kF​)2 in $\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})$ as compared to $\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})$ is a direct consequence of Galilean invariance. The imaginary part of the nematic susceptibility increases linearly with $\ensuremath{\omega}$ up to a peak at the ultraviolet energy scale---the plasma frequency and/or Fermi energy---and then decreases with $\ensuremath{\omega}$. We also obtain explicit forms of the spin susceptibility from the kinetic equation in the collisionless limit and for the Landau function that contains up to the first three harmonics.