BCS-like disorder-driven instabilities and ultraviolet effects in nodal-line semimetals

We study the effects of quenched disorder on electrons in a 3D nodal-line semimetal. Disorder leads to significant renormalisations of the quasiparticle properties due to ultraviolet processes, i.e., processes of scattering in a large band of momenta, of the width exceeding the inverse mean free path. As a result, observables such as the density of states and conductivity exhibit singular behaviour in a broad range of disorder strengths, excluding a small vicinity of the singular point. We find that, for example, the density of quasiparticle states diverges as a function of the disorder strength $g$ as $\rho(g,E)\propto |g_c(E)-g|^{-2}|E|$ for $g$ smaller than the critical value $g_c(E)$ and crosses over to a constant for $g$ very close to $g_c(E)$, where $E$ is the quasiparticle energy. For certain disorder symmetries, a 3D disordered nodal-line semimetal can be mapped to a 2D metal with attractive interactions. The described disorder-driven instabilities in such a nodal-line semimetal are mapped to Cooper and exciton-condensation instabilities in a 2D metal. For other disorder symmetries, the respective instabilities are similar but not exactly dual. We discuss experimental conditions favourable for the observation of the described effects.