= Uniform spin field and magnetization
{title2=$M=U(n_\uparrow-n_\downarrow)=2U\langle S_z\rangle$}
For the auxiliary-field convention with cost $M^2/(2U)$ and coupling $-M\bar\psi\sigma_z\psi$, the homogeneous <saddle point> equation equates M to U times spin polarization per site. M has energy units. It is not itself a magnetic moment, and its coefficient changes if the contact interaction or density-of-states spin convention is changed. The exact three-component decoupling must be distinguished from selecting one constant spin direction as a <mean-field approximation>.
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