 Solution

Let $D_n={\hat{n}}_{n\uparrow }{\hat{n}}_{n\downarrow }$. On a site, empty and doubly occupied states have spin zero, while either singly occupied state has spin $1/2$. ThusThe on-site spin-square normal-ordering identity rewrites the interaction as $3U{\sum }_n{D}_n$. Its contact coefficient is $3U$ in this parameterization, rather than the $U$ convention often used for the Hubbard model.

Using a fermionic coherent-state trace and time slicing, replace each normal-ordered monomial by its Grassmann field symbol. With ${\xi }_{\mathbf{k}}={ϵ}_{\mathbf{k}}-\mu$,The independent fields have antiperiodic thermal boundary conditions. Normal ordering is crucial: the explicit $3\hat{n}/4$ subtraction removes the spin-square contraction, so there is no extra chemical-potential shift from it.