Charge transport near quantum criticality: Exact solution of the Boltzmann equation

We obtain an exact analytical solution of the linearized Boltzmann equation for a non-Galilean-invariant Fermi liquid with disorder, and near a quantum critical point of the Pomeranchuk type (q=0). The solution, expressed in terms of associated Legendre functions, captures the full dependence on the ratio of the impurity and interaction scattering times and provides an exact description of the conductivity over the entire temperature range—from the low-temperature (ballistic) regime, dominated by electron-impurity scattering with time τ_i, to the high-temperature (hydrodynamic) regime, dominated by inelastic scattering with time τ_ee. The crossover between the two regimes occurs when τ_i~τ_ee.

In the Fermi-liquid limit and in two dimensions, the electron-electron scattering rate obeys 1/τ_ee~ξ⁴T²ln(T), leading to σ~ -T⁴lnT and σ~ -T² in the ballistic and hydrodynamic regimes, correspondingly. To extend the theory to the non-Fermi-liquid regime, we incorporate the effects of critical fluctuations phenomenologically, by replacing the correlation length with its critical counterpart, ξ~T^(-1/3), appropriate to a q=0 Pomeranchuk-like instability of Ising-nematic type. This modification changes the inelastic scattering rate to the non-Fermi-liquid form 1/τ_ee~T^2/3,

resulting in σ~ -T^8/3 near criticality. An interesting implication of our result is that a crossver between the ballistic and hydrodynamic regimes occurs at τ_i~τ_ee~T^(-2/3)<<T^(-1), which violates the Planckian bound.