= Miscibility criterion for a two-component condensate
{title2=$U_{11}>0,\quad U_{22}>0,\quad U_{11}U_{22}>U_{12}^2$}
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 $\left(\begin{smallmatrix}U_{11}&U_{12}\\U_{12}&U_{22}\end{smallmatrix}\right)$. 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.
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