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The single-band Hubbard model balances electron hopping between localized spatial orbitals against local interaction. A common form isThe first sum is over neighbouring unordered site pairs. The parameter $t$ sets hopping, while $U_H>0$ penalizes double occupancy. At $U_H=0$ electrons occupy independent energy bands; strong repulsion suppresses local charge fluctuations and can favour local-moment behaviour. This competition was the motivation for the model's treatment of narrow-band electron correlation.^[ref]

At one electron per site and $U_H\gg |t|$, virtual double occupancy yields the antiferromagnetic scale $J_{\mathrm{ex}}=4t^2/U_H$. The two-site Hubbard dimer makes this concrete: the singlet state is lowered by $(\sqrt{U_H^2+16t^2}-U_H)/2$, while triplet states have no such shift. Its small-hopping limit gives Hubbard kinetic exchange.

Interaction conventions must match. For dimensionless spin-half operators, $-2U({\mathbf{S}}^2-3n/4)=3U{n}_{\uparrow }{n}_{\downarrow }$, so that spin-square convention has $U_H=3U$, not $U_H=U$.

A theoretical condensed matter physicist who introduced the Hubbard model in 1963 to study electron correlation in narrow energy bands. The model balances motion of electrons between lattice sites against the Coulomb interaction cost of putting two electrons on the same site. By retaining these competing ingredients, it makes the crossover between mobile and localized electrons accessible without modeling every detail of a real solid.^[ref]

The Hubbard model combines hopping with a local repulsion,Hopping favours a broad itinerant band; the interaction penalizes opposite-spin double occupancy. For uniform Stoner decoupling of the Hubbard interaction, let $s=n_{\uparrow }-n_{\downarrow }$ be the spin imbalance per site. The interaction is $U(n^2-s^2)/4$, so polarization lowers it by $U{s}^2/4$, at the expense of kinetic energy.

If $D(E_F)$ is the noninteracting density of states per spin per site, small polarization costs $s^2/[4D(E_F)]$. Thereforeis the mean-field ferromagnetic instability criterion. The exchange splitting is $\Delta =Us$, and the paramagnetic response is enhanced as $\chi ={\chi }_0/[1-UD(E_F)]$. A density of states including both spins changes the written factor in the criterion.

The model explains how itinerant bands can polarize without imposing pre-existing localized atomic spins. It is not a claim that every large-$U$ metal becomes ferromagnetic: near half filling the Hubbard model can instead form a Mott insulator with antiferromagnetic superexchange of scale $4t^2/U$. Band filling, bandwidth and fluctuations matter.^[ref] The interaction-energy and kinetic-cost expansion above is the stated mean-field derivation of the enhancement.^[ref]