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arXiv:cond-mat/0503718 · Physical Review B

Zero-temperature optical conductivity of ultraclean Fermi liquids and superconductors

Achim Rosch, P. C. Howell

Published Sep 13, 2005• 34 citations
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Abstract

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 $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega}>0)\ensuremath{\approx}\mathrm{const}$ 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 $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega}>0)\ensuremath{\propto}{\ensuremath{\omega}}^{4}$ 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 $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}$.

DOI
10.1103/physrevb.72.104510

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We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface ``hot spots'' and the remainder of the ``cold'' Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, $\ensuremath{\varphi}$, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or $\ensuremath{\varphi}$ fields: These have important effects also away from the hot spots. The simplest composite operator, ${\ensuremath{\varphi}}^{2}$, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.
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