Rotation and Anisotropic Scaling in Axionic Holographic Superconductors: AdS$_4$ versus Lifshitz$_5$

Abstract

An outstanding question in holographic superconductivity is how rotation and non-relativistic scaling symmetries compete to modify the onset of condensation. We address this by comparing the probe superconductor sectors of two exact rotating axionic black holes: a four-dimensional asymptotically anti-de Sitter solution dressed by a non-minimally coupled scalar, and a five-dimensional asymptotically Lifshitz solution with $z \in (2,3)$. Writing both metrics in a common stationary form yields a master scalar equation in which rotation enters through the co-rotating potential $A_t-N^xA_x$. This equation explains why the temporal and spatial components of the probe gauge field cannot be treated independently. The numerical profiles reported in [1,2] indicate that the correlated deformation associated with rotation reduces the condensate amplitude in both backgrounds, whereas increasing the dynamical exponent at fixed $J$ enhances the Lifshitz condensate. Since the parameters $α$ and $J$ also modify the matter fields supporting the corresponding exact geometries, these trends should be interpreted as properties of complete correlated families rather than as universal effects of mechanical rotation alone.