the same as it was for $g_{12}=0$, apart from a slight hardening of the phonon excitations, the spin structure factor \[Fig. \[Fig:DynSF\] (c)\] is qualitatively different. In fact, the spin dispersion relation becomes parabolic. This happens because the spin excitations become effectively noninteracting when $g_{12}=g$ since there is no preference for miscibility over immiscibility.

### BdG energies and the shifted phase transition

We now extend the discussion to beyond the Thomas-Fermi regime to include small $g$, where spin excitations may now experience an effective attractive interaction. The lowest BdG energies as a function of $g_{12}$ are plotted in the first column of Fig. \[Fig:SFandE\], for $g=\{500,8,4\}$, from top to bottom. The phase transition can be identified as where the energy of at least one excitation softens to zero. This occurs at the same point when approaching from either side, consistent with the second-order nature of the phase transition. While in the Thomas-Fermi regime \[Fig. \[Fig:SFandE\](a1)\] the transition occurs at $g_{12}^{\rm c}/g\approx1$, a considerable shift to $g_{12}^{\rm c}/g>1$ can be seen for small $g$ in Figs. \[Fig:SFandE\](b1-c1). The region of effective attractive interactions and enhanced quantum spin fluctuations, $g<g_{12}<g_{12}^{\rm c}$, is schematically shaded with blue (darker in grayscale) color.

### Static 