A deformed IR: a new IR fixed point for four-dimensional holographic theories

In holography, the IR behavior of a quantum system at nonzero density is described by the near horizon geometry of an extremal charged black hole. It is commonly believed that for systems on $S^3$, this near horizon geometry is $AdS_2\times S^3 $. We show that this is not the case: generic static, nonspherical perturbations of $AdS_2\times S^3 $ blow up at the horizon, showing that it is not a stable IR fixed point. We then construct a new near horizon geometry which is invariant under only $SO(3)$ (and not $SO(4)$) symmetry and show that it is stable to $SO(3)$-preserving perturbations (but not in general). We also show that an open set of nonextremal, $SO(3)$-invariant charged black holes develop this new near horizon geometry in the limit $T \to 0$. Our new IR geometry still has $AdS_2$ symmetry, but it is warped over a deformed sphere. We also construct many other near horizon geometries, including some with no rotational symmetries, but expect them all to be unstable IR fixed points.