  Interacting Bose-Fermi mixtures possess a fermionic (super)symmetry when
bosons and fermions in the mixture have equal masses, and when the interaction
strengths are appropriately tuned. This symmetry is spontaneously broken in the
ground state of the mixture, leading to a novel Goldstone mode with fermionic
statistics and quadratic dispersion. Here we examine the effect of explicit
symmetry-breaking perturbations on the Goldstone mode. When the symmetry is not
exact and the system is allowed to deviate from the symmetric point, we find
that the Goldstone mode acquires an energy gap. We show that the excitations
manifest themselves as a non-analyticity of the thermodynamic pressure.
