  We consider a BMS-type symmetry action on isolated horizons in asymptotically
flat spacetimes. From the viewpoint of the non-relativistic field theory on a
horizon membrane, supertranslations shift the field theory spatial momentum.
The latter is related by a Ward identity to the particle number symmetry
current and is spontaneously broken. The corresponding Goldstone boson shifts
the horizon angular momentum and can be detected quantum mechanically.
Similarly, area preserving superrotations are spontaneously broken on the
horizon membrane and we identify the corresponding gapless modes. In
asymptotically AdS spacetimes we study the BMS-type symmetry action on the
horizon in a holographic superfluid dual. We identify the horizon
supertranslation Goldstone boson as the holographic superfluid Goldstone mode.
