Zero-temperature optical conductivity of ultraclean Fermi liquids and superconductors

We calculate the low-frequency optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ of clean metals and superconductors at zero temperature neglecting the effects of impurities and phonons. In general, the frequency and temperature dependences of $\ensuremath{\sigma}$ have very little in common. For small Fermi surfaces in three dimensions (but not in two dimensions) we find, for example, that $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega}g0)\ensuremath{\approx}\mathrm{const}$ which corresponds to a scattering rate $\ensuremath{\Gamma}\ensuremath{\propto}{\ensuremath{\omega}}^{2}$ even in the absence of umklapp scattering when there is no ${T}^{2}$ contribution to $\ensuremath{\Gamma}$. In the main part of the paper we discuss in detail the optical conductivity of $d$-wave superconductors in two dimensions where $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega}g0)\ensuremath{\propto}{\ensuremath{\omega}}^{4}$ for the smallest frequencies and the umklapp processes typically set in smoothly above a finite threshold ${\ensuremath{\omega}}_{0}$ smaller than twice the maximal gap $\ensuremath{\Delta}$. In cases where the nodes are located at $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}∕2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}∕2)$, such that direct umklapp scattering among them is possible, one obtains $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}$.