Fermi–Hubbard Model

The repulsive Fermi–Hubbard model asks how electrons that can hop between sites behave when double occupancy costs energy. It is a minimal setting for the competition between itinerant motion and interaction-driven localization. At half filling and strong repulsion, a controlled virtual-hopping calculation produces antiferromagnetic spin exchange.

Lattice Hamiltonian and state

Take spin-$1/2$ fermions on a two-dimensional square lattice with one orbital per site, nearest-neighbor hopping, and no magnetic flux. The grand-canonical Hamiltonian is

$$\begin{aligned} H={}&-t\sum_{\langle ij\rangle,\sigma} (c_{i\sigma}^{\dagger}c_{j\sigma}+\mathrm{h.c.})\\ &+U\sum_i n_{i\uparrow}n_{i\downarrow} -\mu\sum_{i,\sigma}n_{i\sigma}, \qquad U>0. \end{aligned}$$

The operators anticommute, $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$, $t$ is real, and each unoriented bond is counted once. A site can be empty, singly occupied with either spin, or doubly occupied. Boundary conditions, temperature, and filling complete the problem; half filling means one fermion per site on average.

This Hamiltonian conserves total particle number and has spin $SU(2)$ symmetry. It defines a lattice quantum many-body theory, rather than a relativistic continuum theory at all scales. Other lattices, longer-range hopping, and additional orbitals are distinct variants.

Virtual hopping and spin exchange

At half filling with $U\gg |t|$, states with one fermion on every site have no double-occupancy cost. One hop out of this subspace creates an empty site and a doubly occupied site, costing $U$. A second hop can return to the original configuration or exchange the two spins.

For one bond, second-order perturbation theory lowers the singlet energy by $4t^2/U$ while leaving the triplet unshifted. These eigenvalues are reproduced by

$$H_{\mathrm{spin}} =J\sum_{\langle ij\rangle} \left(\mathbf S_i\cdot\mathbf S_j-\frac14\right), \qquad J=\frac{4t^2}{U}>0,$$

within the singly occupied subspace. Indeed, $\mathbf S_i\cdot\mathbf S_j$ is $-3/4$ for a singlet and $1/4$ for a triplet. The leading spin Hamiltonian therefore favors neighboring spins in a singlet, as described in Arovas et al. 2021, §5.2.

For an isolated two-site system with two fermions, the lower singlet energy relative to the singly occupied atomic energy is exactly

$$E_s=\frac{U-\sqrt{U^2+16t^2}}2 =-\frac{4t^2}{U}+O(t^4/U^3).$$

This gives a direct check of the factor four and the antiferromagnetic sign.

Scales, observables, and limits

Charge fluctuations cost an energy of order $U$ in this regime, while spin dynamics occurs on the much smaller scale $J$. A temperature small compared with $U$ need not be small compared with $J$. Local moment formation and low-temperature spin correlations are consequently separate questions.

Useful observables include double occupancy, density response, spin correlations, and the single-particle spectral function. The existence of local antiferromagnetic exchange does not by itself establish long-range order: dimension, temperature, and lattice geometry matter. Doping restores mobile charge carriers and requires more than the half-filled spin-only Hamiltonian.

At $U=0$ the model reduces to a hopping band; at $t=0$ it becomes independent interacting sites. Neither limit solves the general intermediate-coupling problem, and fitting a real material to this one-band model requires a separate justification.

Develop the calculation

The Hubbard Model and Mott Regime develops the separation of spin and charge scales. Compare the on-site occupation problem for bosons in the Bose–Hubbard Model.

References

• Arovas, Daniel P., Erez Berg, Steven A. Kivelson, and Srinivas Raghu. The Hubbard Model. Preprint (2021), arXiv:2103.12097v1. Open text.