  We observe a condensate forming in the interior of a black hole (BH) during
numerical simulations of gravitational collapse of a massless charged (complex)
scalar field. The magnitude of the scalar field in the interior tends to a
non-zero constant; spontaneous breaking of gauge symmetry occurs and a
condensate forms. This phenomena occurs in the presence of a BH without the
standard symmetry breaking quartic potential; the breaking occurs via the
dynamics of the system itself. We also observe that the scalar field in the
interior rotates in the complex plane and show that it matches numerically the
electric potential to within $1\%$. That a charged scalar condensate can form
near the horizon of a black hole in the Abelian Higgs model without the
standard symmetry breaking potential had previously been shown analytically in
an explicit model involving a massive scalar field in an $AdS_4$ background.
Our numerical simulation lends strong support to this finding, although in our
case the scalar field is massless and the spacetime is asymptotically flat.
