= Solution
In the homogeneous spin-field sector, choose $\mathbf M=M\hat{\mathbf e}_z$. The fermionic levels become $\epsilon_{\mathbf k}-\mu-\sigma M$, where $\sigma=\pm1$ is the Pauli eigenvalue. The <fermionic one-level thermal determinant> can be applied independently to every momentum and spin, while the field cost is $\beta NM^2/(2U)$. Hence the <homogeneous spin-field effective action> is
$$
\boxed{\mathcal S_{\rm eff}[M]=-\sum_{\mathbf k,\sigma}\ln[1+e^{-\beta(\epsilon_{\mathbf k}-\mu-\sigma M)}]+\frac{\beta NM^2}{2U}.}
$$
The remaining homogeneous-field integral is $\mathcal Z\simeq\int\mathcal DM\,e^{-\mathcal S_{\rm eff}}$ 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.
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