Conductivity of a Non-Galilean--Invariant Fermi Liquid: Exact Solution of the Kinetic Equation

TL;DR

This paper provides an exact solution for the conductivity of a disordered, non-Galilean-invariant Fermi liquid, revealing how electron-electron interactions influence conductivity through the quasiparticle scattering time and detailing regime crossovers.

Contribution

It presents an exact kinetic equation solution incorporating Coulomb and Pomeranchuk interactions, clarifying the role of electron-electron interactions and mass renormalization in conductivity.

Findings

Electron-electron interactions affect conductivity solely via quasiparticle scattering time.

Crossover between collisionless and hydrodynamic regimes occurs when $1/\tau_{ee}$ matches impurity rate or probe frequency.

Hydrodynamic optical response is derived, inaccessible by perturbation theory.

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, $\tau_\mathrm{ee}$. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when $1/\tau_\mathrm{ee}$ becomes comparable to the larger of the impurity scattering rate and the probe frequency, $\Omega$. In addition, the exact solution yields the optical response in the hydrodynamic regime, $\Omega\ll 1/\tau_\mathrm{ee}$, 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 $\Omega\sim T$.