Instabilities in a Two-Component, Species Conserving Condensate

We consider a system of two species of bosons of equal mass, with interactions $U^{a}(|x|)$ and $U^{x}(|x|)$ for bosons of the same and different species respectively. We present a rigorous proof -- valid when the Hamiltonian does not include a species switching term -- showing that, when $U^{x}(|x|)>U^{a}(|x|)$, the ground state is fully "polarized" (consists of atoms of one kind only). In the unpolarized phase the low energy excitation spectrum corresponds to two linearly dispersing modes that are even a nd odd under species exchange. The polarization instability is signaled by the vani shing of the velocity of the odd modes.