Holographic Superconductors with Lifshitz Scaling

TL;DR

This paper develops a holographic framework for superconductivity at Lifshitz fixed points with $z>1$ by extending the Lifshitz gravity model with a Maxwell field and a charged scalar. It analyzes both exact ($z=2,4$) and numerical Lifshitz black holes, showing a universal null curvature singularity at $r=0$ and constructing charged, hairy Lifshitz black holes that undergo a superconducting phase transition as temperature decreases. A Lifshitz analog of AdS-RN is recovered at $z=1$, while for general $z>1$ the authors demonstrate scalar hair formation and boundary operator condensation, signaling superconductivity in the dual theory. The work provides a viable route to study finite-temperature, finite-density strongly coupled systems with anisotropic scaling and may illuminate quantum critical behavior beyond relativistic holography.

Abstract

Black holes in asymptotically Lifshitz spacetime provide a window onto finite temperature effects in strongly coupled Lifshitz models. We add a Maxwell gauge field and charged matter to a recently proposed gravity dual of 2+1 dimensional Lifshitz theory. This gives rise to charged black holes with scalar hair, which correspond to the superconducting phase of holographic superconductors with z > 1 Lifshitz scaling. Along the way we analyze the global geometry of static, asymptotically Lifshitz black holes at arbitrary critical exponent z > 1. In all known exact solutions there is a null curvature singularity in the black hole region, and, by a general argument, the same applies to generic Lifshitz black holes.