Homogeneous spin-field effective action ($S_{\mathrm{eff}}=\beta N{M}^2/(2U)-{\sum }_{k\sigma }\ln [1+e^{-\beta ({ϵ}_k-\mu -\sigma M)}]$)

Integrating free fermions in a constant auxiliary spin field gives $S_{\mathrm{eff}}=\beta N{M}^2/(2U)-{\sum }_{k\sigma }\ln [1+e^{-\beta ({ϵ}_k-\mu -\sigma M)}]$. Differentiation gives the self-consistent spin polarization. The extensive action is a mean-field restriction of the full auxiliary-field functional integral, not an exact omission of all spatial/time fluctuations.