The Hubbard model and Mott localization

Take one orbital per site, two spin states, nearest-neighbor hopping $t>0$, and repulsion $U>0$. The local occupations are empty, up, down, and doubly occupied. Hopping favors states spread over the lattice, whereas the interaction penalizes double occupancy. The model omits other orbitals, long-range Coulomb terms, lattice vibrations, and disorder. Those omissions can matter for a material, so a successful qualitative mechanism is not automatically a quantitative fit.

Half filling means one electron per site on average, not a prohibition against empty or doubly occupied sites. At $U=0$, the half-filled band can conduct. At $t=0$, moving charge between two singly occupied sites creates one empty site and one double occupation at cost $U$. Spin orientations remain degenerate in this atomic limit. Charge localization therefore does not require the noninteracting band to be full, and it does not by itself choose a magnetic pattern.

At half filling with $U\gg t$, low-energy configurations have almost one particle per site. A single hop leaves that subspace, but a second hop can return to it. The intermediate double occupation is virtual: it contributes to an amplitude without becoming a long-lived low-energy charge excitation. Second-order perturbation theory produces an exchange proportional to $t^2/U$. This expansion requires the charge-excitation denominator to remain large compared with the hopping.

For the two-site problem, the spin singlet can lower its energy by admixing doubly occupied configurations; the triplet does not gain the same energy. The resulting exchange is antiferromagnetic. After an additive constant, the effective Hamiltonian is a Heisenberg spin model. Its spin dynamics occur on a scale much smaller than the charge penalty. Deriving that scale is controlled at small $t/U$; solving a frustrated spin model at arbitrary size can still be difficult.

A band insulator can arise without interactions when all available low-energy bands are filled. A Mott insulator instead involves suppressed charge motion caused by interactions in a setting where band counting alone would allow a metal. In practice, magnetic order can also reconstruct bands. Separating these mechanisms requires attention to symmetry, temperature, spectral gaps, and how the state evolves as parameters change. The label should summarize evidence rather than replace it.

Away from half filling, holes can move without always creating the same expensive double occupation, and the spin-only description no longer contains all low-energy degrees of freedom. Intermediate coupling in two dimensions remains a demanding many-body problem. Mean-field, finite-cluster, Monte Carlo, and tensor-network calculations have distinct errors and accessible regimes. Agreement in a controlled limit is a useful calibration; it cannot settle every parameter region that the approximation does not control.