Zero temperature optical conductivity of ultra-clean Fermi liquids and superconductors

Abstract

We calculate the low-frequency optical conductivity sigma(w) of clean metals and superconductors at zero temperature neglecting the effects of impurities and phonons. In general, the frequency and temperature dependences of sigma have very little in common. For small Fermi surfaces in three dimensions (but not in 2D) we find for example that Re sigma(w>0)=const. for low w which corresponds to a scattering rate Gamma proportional to w^2 even in the absence of Umklapp scattering when there is no T^2 contribution to Gamma. In the main part of the paper we discuss in detail the optical conductivity of d-wave superconductors in 2D where Re sigma(w>0) \propto w^4 for the smallest frequencies and the Umklapp processes typically set in smoothly above a finite threshold w_0 smaller than twice the maximal gap Delta. In cases where the nodes are located at (pi/2, pi/2), such that direct Umklapp scattering among them is possible, one obtains Re sigma(w) \propto w^2.