 Miscibility criterion for a two-component condensate

In a homogeneous overlapping two-component Bose-Einstein condensate, the long-wavelength energy change at fixed species numbers is one half the integral of the density perturbation quadratic form with matrix $\left(\begin{array}{cc}{U}_{11}& U_{12}\\ U_{12}& U_{22}\end{array}\right)$. Strict positive definiteness gives the displayed criterion. Strong positive cross-repulsion favors separation; sufficiently strong cross-attraction instead produces an unstable joint density mode. Equality is marginal at this order, not strict stability.