Three-point density correlations in a weakly interacting 2D Fermi liquid

We study the three-point equal-time correlations of the density in a weakly interacting spin 1/2 Fermi gas and present two new results. First, we compute the three-point correlation for the total density $\rho = \rho_\uparrow + \rho_\downarrow$ exactly as a function of momentum to first order in a dimensionless interaction parameter ${\cal I}$. This generalizes a previous result that related the three-point function to the Landau Fermi liquid parameters $F_0^s$ and $F_0^a$ and applied in a certain long-wavelength collinear limit. Second, we compute the leading order ${\cal O}({\cal I}^3)$ interaction correction to the same-spin three-point correlation function in the long-wavelength collinear limit. These results are directly relevant to current experiments on atomic Fermi gases using quantum gas microscopy.