Transport in the non-Fermi liquid phase of isotropic Luttinger semimetals

Abstract

Luttinger semimetals have quadratic band crossings at the Brillouin-zone center in three spatial dimensions. Coulomb interactions in a model that describes these systems stabilize a nontrivial fixed point associated with a non-Fermi liquid state, also known as the Luttinger-Abrikosov-Beneslavskii phase. We calculate the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and the dc conductivity ${\ensuremath{\sigma}}_{dc}(T)$ of this phase, by means of the Kubo formula and the Mori-Zwanzig memory matrix method, respectively. Interestingly, we find that $\ensuremath{\sigma}(\ensuremath{\omega})$, as a function of the frequency $\ensuremath{\omega}$ of an applied ac electric field, is characterized by a small violation of the hyperscaling property in the clean limit, which is in contrast with the low-energy effective theories that possess Dirac quasiparticles in the excitation spectrum and obey hyperscaling. Furthermore, the effects of weak short-ranged disorder on the temperature dependence of ${\ensuremath{\sigma}}_{dc}(T)$ give rise to a stronger power-law suppression at low temperatures compared to the clean limit. Our findings demonstrate that these disordered systems are actually power-law insulators. Our theoretical results agree qualitatively with the data from recent experiments performed on Luttinger semimetal compounds like the pyrochlore iridates $[{({\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{Pr}}_{x})}_{2}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}]$.