 Solution

The Hubbard model combines band motion and local repulsion: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 $U{g}_{\uparrow }(E_F)>1$ 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.