Criteria For Superfluid Instabilities of Geometries with Hyperscaling Violation

We examine the onset of superfluid instabilities for geometries that exhibit hyperscaling violation and Lifshitz-like scaling at infrared and intermediate energy scales, and approach AdS in the ultraviolet. In particular, we are interested in the role of a non-trivial coupling between the neutral scalar supporting the scaling regime, and the (charged) complex scalar which condenses. The analysis focuses exclusively on unstable modes arising from the hyperscaling-violating portion of the geometry. Working at zero temperature, we identify simple analytical criteria for the presence of scalar instabilities, and discuss under which conditions a minimal charge will be needed to trigger a transition. Finite temperature examples are constructed numerically for a few illustrative cases.