Geometry meets interactions: how the shape of the Fermi surface impacts the dynamics and quantum critical behavior of 2D electrons

We demonstrate that the optical conductivity of a Fermi liquid (FL) in the absence of umklapp scattering is dramatically affected by the geometry and topology of the Fermi surface (FS). Specifically, electron-electron (ee) interaction leads to rapid current relaxation in systems with multiple, or multiply connected, FSs, provided the valleys have different effective masses. This effect results from intervalley drag. Similar result holds for the concave Fermi surface. We microscopically derive the optical conductivity, 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, Re σ(ω)∼const. This dependence contrasts sharply with the sub-leading contribution to the conductivity of a two-dimensional FL with a single convex FS, where Re σ(ω)∼ω^2 ln|ω|. The vanishing of the leading term in the optical conductivity is a signature of geometric constraints on ee scattering channels, which are lifted both for multiply connected and concave FSs. On the verge of the topological Lifshitz transition from a single-valley to a multi-valley FS, giant differential conductivity is predicted, giving rise to a distinct experimental footprint within the experimentally accessible frequency range. Near a QCP, intervalley drag leads to a |ω|^{−2/3} scaling of Re σ(ω) in 2D, thus providing a specific current-relaxing process for this long-standing conjecture.