= Solution
The <Hubbard model> combines band motion and local repulsion:
$$
H=-t\sum_{\langle ij\rangle,\sigma}(c_{i\sigma}^\dagger c_{j\sigma}+\mathrm{h.c.})+U\sum_i n_{i\uparrow}n_{i\downarrow}.
$$
Hopping favors itinerant states, while positive $U$ penalizes simultaneous opposite-spin occupation on one site. In a uniform mean-field treatment, a spin imbalance reduces the interaction cost but raises the band-filling <energy>. The <Stoner criterion> for instability of the unpolarized metal is $Ug_\uparrow(E_F)>1$ when $g_\uparrow$ is the density of states per site, per <energy>, for one spin. Using a total two-spin density of states changes the displayed numerical convention.
The ordered mean-field state has exchange-split spin bands and a spontaneous magnetization, with the kinetic and interaction costs balancing at equilibrium. This is a model of itinerant <ferromagnetism>, not a guarantee that every repulsive Hubbard system becomes ferromagnetic. Filling, lattice structure and competing order matter: near half filling on a bipartite lattice, antiferromagnetism and a <Mott insulator> can be favored instead. The approximation also omits important spin fluctuations and strong-correlation effects.
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