  We numerically calculate the density profile and excitation spectrum of a
two-species Bose-Einstein condensate for the parameters of recent experiments.
We find that the ground state density profile of this system becomes unstable
in certain parameter regimes, which leads to a phase transition to a new stable
state. This state displays spontaneously broken cylindrical symmetry. This
behavior is reflected in the excitation spectrum: as we approach the phase
transition point, the lowest excitation frequency goes to zero, indicating the
onset of instability in the density profile. Following the phase transition,
this frequency rises again.
