Hubbard model ($H=H_{\mathrm{hop}}+U_H{\sum }_n{n}_{n\uparrow }{n}_{n\downarrow }$)

A lattice fermion model combining hopping with local density interaction, conventionally written $U_H{\sum }_n{n}_{n\uparrow }{n}_{n\downarrow }$. Its interaction parameter must be matched when a spin-square representation is used: $-2U({\mathbf{S}}^2-3n/4)$ equals $3U{n}_{\uparrow }{n}_{\downarrow }$, so $U_H=3U$ for that representation.

For a single spinful fermion orbital, ${\mathbf{S}}^2=3(n-2n_{\uparrow }{n}_{\downarrow })/4$. The contraction term is $3n/4$, so $:{\mathbf{S}}^2:={\mathbf{S}}^2-3n/4=-3n_{\uparrow }{n}_{\downarrow }/2$. Empty and doubly occupied states are spinless while singly occupied states have spin one half, which also proves the identity directly.

Ferromagnetic spin polarization carried by mobile electrons rather than a separate fixed-spin lattice. A homogeneous spin field splits the electron bands and competes with a positive field cost. Mean-field instability is governed by the Stoner criterion; the ordered metal supports transverse spin waves as well as fermionic particle-hole excitations.

An isotropic spin-polarized metal has a degenerate magnetization direction and a long-wavelength transverse magnon with quadratic dispersion. Nonzero spin density pairs the two transverse broken generators into one type-B Goldstone boson. Exchange-split Fermi surfaces also support fermionic low-energy excitations; the magnon is distinct from longitudinal amplitude motion and the spin-flip continuum.

In the homogeneous spin-vector mean-field scheme with field cost $M^2/(2U)$ and spin-summed density of states per site, the paramagnetic curvature is proportional to $U^{-1}-\int \nu (ϵ)[-n_F^{\prime }(ϵ)]dϵ$. It changes sign at $U{\chi }_0=1$. For a smooth low-temperature density of states, ${\chi }_0\simeq \nu (\mu )$, giving the familiar Fermi-level criterion. Interaction and spin-counting conventions must be specified before comparing coefficients.

Integrating free fermions in a constant auxiliary spin field gives $S_{\mathrm{eff}}=\beta N{M}^2/(2U)-{\sum }_{k\sigma }\ln [1+e^{-\beta ({ϵ}_k-\mu -\sigma M)}]$. Differentiation gives the self-consistent spin polarization. The extensive action is a mean-field restriction of the full auxiliary-field functional integral, not an exact omission of all spatial/time fluctuations.