Fermi surface geometry and optical conductivity of a two-dimensional electron gas near an Ising-nematic quantum critical point

We analyze optical conductivity of a clean two-dimensional electron system in a Fermi liquid regime near a $T=0$ Ising-nematic quantum critical point (QCP) and extrapolate the results to a QCP. We employ direct perturbation theory up to the two-loop order to elucidate how the Fermi surface's geometry (convex vs concave) and fermionic dispersion (parabolic vs nonparabolic) affect the scaling of the optical conductivity σ(ω) with frequency ω and correlation length ξ. We find that for a convex Fermi surface the leading terms in the optical conductivity cancel out, leaving a subleading contribution σ(ω)∝ω²ξ⁴L, where L=const for a parabolic dispersion and L∝lnωξ³ in a generic case. For a concave Fermi surface, the leading terms do not cancel, and σ(ω)∝ξ². We extrapolate these results to a QCP and obtain σ(ω)∝ω2/3 for a convex Fermi surface and σ(ω)∝1/ω2/3 for a concave Fermi surface.