Spontaneous scalarization around a black hole in a dark matter halo

We demonstrate that in a scalar-tensor theory where dark matter is conformally coupled to a scalar field, a black hole surrounded by a Hernquist dark matter halo can undergo spontaneous scalarization, acquiring a nontrivial scalar field configuration. The astrophysical properties of the dark matter halo directly dictate the quantum-like spectrum of scalar field configurations around the central black hole. Remarkably, the conformal coupling constant becomes discretized and is determined solely by the halo's compactness parameter.