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

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 $Ω\sim T$.