Quantum criticality and optical conductivity in a two-valley system

We demonstrate that the optical conductivity of a Fermi liquid (FL) in the absence of umklapp scattering is dramatically affected by the topology of the Fermi surface (FS). Specifically, electron-electron ($\mathit{ee}$) scattering leads to rapid current relaxation in systems with multiple, or multiply connected, FSs, provided that the valleys have different effective masses. This effect results from intervalley drag. We microscopically derive the optical conductivity of a two-valley system, both within the FL regime and near a quantum critical point (QCP) of the Ising-nematic type. In the FL regime, intervalley drag restores the Gurzhi-like scaling of the conductivity, $\mathrm{Re}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{0}$. This dependence contrasts sharply with the previously identified subleading contribution to the conductivity of a two-dimensional FL with a single convex FS, where $\mathrm{Re}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2}ln|\ensuremath{\omega}|$. The vanishing of the leading term in the optical conductivity is a signature of geometric constraints on $\mathit{ee}$ scattering channels, which are lifted for a multiply connected FS. A large differential response, $d\mathrm{Re}\ensuremath{\sigma}/d\ensuremath{\mu}$ with $\ensuremath{\mu}$ being the chemical potential, is predicted at the Lifshitz transition from a single-valley to a multivalley FS, which should be observable within the experimentally accessible frequency range. Near a QCP, intervalley drag leads to a ${|\ensuremath{\omega}|}^{\ensuremath{-}2/3}$ scaling of $\mathrm{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ in two dimensions, thus providing a specific current-relaxing process for this long-standing conjecture.