Equal-spin and opposite-spin density-density correlations in the BCS-BEC crossover: Gauge Symmetry, Pauli Exclusion Principle, Wick's Theorem and Experiments

Abstract

We develop a general theory of spin-dependent density-density correlations, that is valid for any temperature, interactions, dimensions and mass or population status of Fermi gases with two internal states. We use gauge invariance and the Pauli principle to establish constraints on the spin-dependent density-density correlations that are consistent with the fluctuation-dissipation and Wick's theorem. As an example, we study the spin-dependent density-density correlations from the BCS to the Bose regime in two dimensions at zero temperature, inspired by experiments in 6Li. We show that two-particle irreducible contributions involving collective excitations, many-particle scattering and vertex corrections, are essential to describe experiments. In particular they turn out to be responsible for the emergence of an experimentally observed minimum in the opposite-spin density-density correlations.

Figures (5)

• Figure 1: Plots of spatial behavior of the correlation functions $g_{\rm s s^\prime} (\delta {\bf r})$ and of $g_{nn} (\delta {\bf r})$ for translationally and rotationally invariant systems. The interaction parameters are $\ln k_{\rm F} a = 2.15$ (BCS regime) and $\ln k_{\rm F} a = 0.36$ (crossover region). The line types indicate the level of approximation used, see Table \ref{['tab:comparison-of-approaches']}.
• Figure 2: Numerical results for $g_{\uparrow\uparrow} (\delta {\bf r})$ ranging from $0$ and $1$ (top panel), $g_{\uparrow\downarrow} (\delta {\bf r})$ (middle panel) ranging from $0.7$ to $1.4$ and $g_{nn} (\delta {\bf r})$ (bottom panel) ranging from $0.5$ to $1.5$ versus distance $k_{\rm F} \vert \delta {\bf r}\vert$ ranging from $0$ to $5$. The dashed black (solid blue) lines illustrate case ${\rm I}$ (IV) in Table \ref{['tab:comparison-of-approaches']}. The gray region in the middle panel shows that $g_{\uparrow\downarrow} (\delta {\bf r}) \ge 1$ with no minimum for the dashed black line, while $g_{\uparrow\downarrow} (\delta {\bf r})$ has a clear minimum below 1 for the solid blue line. The interaction parameter $\ln k_{\rm F} a$ ranges from $2.15$ (BCS regime) to $0.36$ (crossover region) and are closely related to the experimental values Yefsah-2025-3.
• Figure 3: Plots of $k_{\rm F} r_{\rm p}$ (top panel), $k_{\rm F} \vert \delta {\bf r} \vert_{\rm min}$ (middle panel), $h_{\rm min}$ (bottom panel), versus $\ln k_{\rm F} a$. The depth of the minimum is $h_{\rm min} = \vert g_{\uparrow\downarrow} (\delta{\bf r}_{\rm min}) - 1 \vert$ or $h_{\rm min} = \vert g_{nn} (\delta{\bf r}_{\rm min}) - 1 \vert$. The solid blue lines refer to $g_{\uparrow\downarrow} (\delta {\bf r})$, the dotted blue lines represent $g_{\uparrow\downarrow} (\delta {\bf r})$, and the dashed blue line reflects $g_{\uparrow\uparrow} (\delta {\bf r})$. The blue circles, diamonds and squares describe the scattering parameters used in Fig. \ref{['fig:correlation-functions-pauli-preserving']}.
• Figure 4: Self-consistent results for $|\Delta_0|/\varepsilon_{\rm F}$ (top panel) and $\mu/\varepsilon_{\rm F}$ (bottom panel) versus $\ln k_{\rm F}a$ (lower $x$-axis) or binding energy $\varepsilon_{\rm B}/\varepsilon_{\rm F}$ (upper $x$-axis). The dashed black line shows the saddle-point (SP), the dash-dotted red line describes the Gaussian fluctuations (GF), while the solid blue line represents the Pauli-respecting (PR) equation of state (EoS).
• Figure 5: Numerical results for $g_{\uparrow\uparrow} (\delta {\bf r})$ (top panel), $g_{\uparrow\downarrow} (\delta {\bf r})$ (middle panel) and $g_{nn} (\delta {\bf r})$ (bottom panel) versus $k_F \vert \delta {\bf r}\vert$. The dashed black (solid blue) lines illustrate case ${\rm I}$ (IV) in table \ref{['tab:comparison-of-approaches']}. We use $\ln k_{\rm F} a$ from $3.01$ (BCS regime) to $-0.46$ (BEC region).