Self-similar evaporation and collapse in the quantum portrait of black holes

We investigate Hawking evaporation in a recently suggested picture in which black holes are Bose condensates of gravitons at a quantum critical point. There, evaporation of a black hole is due to two intertwined effects. Coherent excitation of a tachyonic breathing mode is responsible for the collapse of the condensate, while incoherent scattering of gravitons leads to Hawking radiation. To explore this, we consider a toy model of a single bosonic degree of freedom with derivative self-interactions. We consider the real-time evolution of a condensate and derive evaporation laws for two possible decay mechanisms in the Schwinger-Keldysh formalism. We show that semiclassical results can be reproduced if the decay is due to an effective two-body process, while the existence of a three-body channel would imply very short lifetimes for the condensate. In either case, we uncover the existence of scaling solutions in which the condensate is at a critical point throughout the collapse. In the case of a two-body decay we moreover discover solutions that exhibit the kind of instability that was recently conjectured to be responsible for fast scrambling.