and the immiscible phase for (b) $g_{12}/g_{12}^{\rm c}=1.1$ and (c) $g_{12}/g_{12}^{\rm c}=1.5$, where $g_{12}^{\rm c}$ is the immiscibility critical point. The first component ($\alpha=1$) is shown in red (gray) while the second component ($\alpha=2$) is black. Column 1 offers a side view ($y=0$) while column 2 consists of top-down views, each exhibiting a single contour at $1/e$ of the peak density. The same two intracomponent interaction strengths are compared in all panels, i.e. $g=\{4,500\}$. The variational predictions $n_\alpha^\mathcal{V}$, shown as thin blue lines, are only plotted for the second component to avoid cluttering. The mixture is balanced, i.e. $g_{11}=g_{22}\equiv g$ and $N_1=N_2$, where $g_{\alpha\beta}=N_\beta\sqrt{8\pi}a_{\alpha\beta}/a_z$, with $N_\beta$ being the number of atoms in component $\beta$, $a_{\alpha\beta}$ is the s-wave scattering length between components $\alpha$ and $\beta$, and $a_{\{\rho,z\}}=\sqrt{\hbar/m\omega_{\{\rho,z\}}}$. \[Fig:DenProfs\]](Fig1){width="3.4in"}

Results
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The Immiscibility Phase Transition {#Sec:ResultsPT}
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In this section we investigate how the immiscible critical value $g_{12}^{\rm c}$ shifts to higher values than the thermodynamic-limit prediction $g_{12}^{\rm c,TD}=g$. As will be shown in Sec. \[Sec:ResultsSF\], when the intercomponent interaction strength lies within the resulting gap, i.e. $g_{12}^{\rm c,TD}<g_{12}<g_{12}^{\rm c}$, then the quantum spin fluctuations become greatly enhanced.

In Fig. \[Fig:DenProfs\], we compare our numerical and variational predictions for the density, 