2}$ )

A Gaussian auxiliary-field identity replacing a quadratic expression in a composite field by a linear coupling to a new field. Completing the square in

M^{2} / (2 U) - M j

yields an exponential

e^{U j^{2} /2}

after integrating

M

, with normalized Gaussian prefactors. It converts suitable quartic field interactions into bilinear couplings and permits fermionic fields to be integrated out.

Table of contents
- Spin-vector Hubbard–Stratonovich decoupling Hubbard–Stratonovich transformation
  - Uniform spin field and magnetization Spin-vector Hubbard–Stratonovich decoupling

Spin-vector Hubbard–Stratonovich decoupling ( $j^{2} = - 6 \overset{ˉ}{ψ}_{↑} \overset{ˉ}{ψ}_{↓} ψ_{↓} ψ_{↑}$ )

 0  0

For two Grassmann spin components,

j = \overset{ˉ}{ψ} σ ψ

satisfies

j^{2} = - 6 \overset{ˉ}{ψ}_{↑} \overset{ˉ}{ψ}_{↓} ψ_{↓} ψ_{↑}

. A real vector auxiliary field with cost

M^{2} / (2 U)

and coupling

- M \cdot j

therefore reproduces the repulsive

3 U

contact action. All three spin components are essential to that normalization.

Uniform spin field and magnetization ( $M = U (n_{↑} - n_{↓}) = 2 U ⟨ S_{z} ⟩$ )

 0  0

For the auxiliary-field convention with cost

M^{2} / (2 U)

and coupling

- M \overset{ˉ}{ψ} σ_{z} ψ

, 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.