structure factor: enhanced quantum fluctuations

Before proceeding, it is worth discussing the relationship between the structure factor and the sign of the interaction for a single-component, uniform system. Consider the Bijl-Feynman formula $S(\mathbf{k})=\epsilon_0(\mathbf{k})/\epsilon_B(\mathbf{k})$, where $\epsilon_0$ is the noninteracting dispersion relation and $\epsilon_B$ is the BdG energy [@Brunello2001]. If an excitation with some $\mathbf{k}$ experiences an effective attractive interaction, then the energy lies below its noninteracting counterpart, thus lifting $S(\mathbf{k})$ above unity as a result of atom bunching with respect to the noninteracting limit.

We present the numerical results for the spin (Fig. \[Fig:SFandE\], column 2) and density (Fig. \[Fig:SFandE\], column 3) static structure factors, restricting our attention to the miscible phase. Consistent with the energy results of the first column, each row in Fig. \[Fig:SFandE\] considers one of $g=\{500,8,4\}$, from top to bottom. The different curves are for various $g_{12}$ and the line styles are chosen to match (within each row) the values marked by vertical lines in column 1. As a benchmark, consider the uncoupled case, $g_{12}=0$, indicated by the green dashed lines. Here, as expected, the spin $S_\mathcal{S}$ and density $S_\mathcal{D}$ structure factors are identical for a given $g$. Increasing $g_{12}$ from zero has the opposite effect on 