Hubbard Model

The Hubbard model is a lattice many-body model in which particles hop between sites and interact locally. The spinful fermionic version is a central minimal model for correlated electrons, Mott physics, magnetism, and lattice many-body methods.

Formula Hook

For spinful fermions on a lattice, a common form is

$$H_{\mathrm{Hub}} = - t \sum_{\langle r,s\rangle,\sigma} \left( c_{r\sigma}^\dagger c_{s\sigma} + c_{s\sigma}^\dagger c_{r\sigma} \right) + U\sum_r n_{r\uparrow}n_{r\downarrow}.$$

The hopping term is one-body. The on-site interaction is two-body because it assigns energy to opposite-spin pairs occupying the same site.

Canonical Home

The canonical model treatment is Hubbard Model. The Hubbard Dimer dossier is the complete two-site record, and the Hubbard Chain dossier is the convention-complete one-dimensional integrable record. For the general second-quantized Hamiltonian context, see Many-Particle Hamiltonians, Two-Body Operators, and Fock Space Examples. The one-particle two-site seed is Tight-Binding Dimer, while Effective Hamiltonians in Many-Body Systems owns the controlled lattice strong-coupling projection.

Common Confusions

• Site labels are mode labels, not labels of distinguishable particles.
• The sign and normalization of $t$ are convention-dependent.
• The ratio $U/t$, filling, dimensionality, lattice geometry, boundary conditions, and spin content must be specified.
• The Hubbard model is minimal, not automatically realistic for every correlated material.

Related Entries

• Hamiltonian
• Density Matrix
• Quantum Hall Effect

References

• J. Hubbard, “Electron correlations in narrow energy bands,” Proceedings of the Royal Society A 276, 238-257, 1963.
• M. C. Gutzwiller, “Effect of correlation on the ferromagnetism of transition metals,” Physical Review Letters 10, 159-162, 1963.
• A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer, 1994.