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Define ${\mathbf{j}}_n={\bar{\psi }}_n\mathbf{\sigma }{\psi }_n$ and $D={\bar{\psi }}_{\uparrow }{\bar{\psi }}_{\downarrow }{\psi }_{\downarrow }{\psi }_{\uparrow }$. The Pauli matrix completeness identity, or direct evaluation of its three components, gives ${\mathbf{j}}^2=-6D$. Consequently the quartic action is $3UD=-U{\mathbf{j}}^2/2$.

The Hubbard–Stratonovich decoupling follows by completing a three-component square at each site and time slice:The constant $C$ is absorbed into the normalized auxiliary-field measure, and $U>0$ ensures convergence of the real Gaussian integral. Thus the spin-vector Hubbard–Stratonovich decoupling yieldsIntegrating out $\mathbf{M}$ reconstructs the normal-ordered interaction exactly. The auxiliary field is periodic in imaginary time, while the fermionic fields remain antiperiodic.

The identity follows because the identity matrix and the three Pauli matrices form an orthogonal basis of two-by-two matrices under the trace inner product. It rewrites a spin-vector bilinear square as a scalar four-fermion expression, with the Grassmann signs supplied when fields are reordered. This is the algebra behind a spin-vector Hubbard–Stratonovich decoupling.