 Solution

In the homogeneous spin-field sector, choose $\mathbf{M}=M{\hat{\mathbf{e}}}_z$. The fermionic levels become ${ϵ}_{\mathbf{k}}-\mu -\sigma M$, where $\sigma =\pm 1$ is the Pauli eigenvalue. The fermionic one-level thermal determinant can be applied independently to every momentum and spin, while the field cost is $\beta N{M}^2/(2U)$. Hence the homogeneous spin-field effective action isThe remaining homogeneous-field integral is $\mathcal{Z}\simeq \int \mathcal{D}M{e}^{-{\mathcal{S}}_{\mathrm{eff}}}$ in this mean-field restriction. Orientation and fluctuation-independent measure factors do not alter the displayed extensive action or its stationary equation. The two bands are exchange split by $2M$; $M$ has energy units and is proportional to the spin polarization, rather than being an electron number itself.