Hubbard Model Hamiltonian

Hamiltonian

A standard spinful fermionic Hubbard Hamiltonian is

$$H = -t \sum_{\langle i,j\rangle,\sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + c_{j\sigma}^\dagger c_{i\sigma} \right) + U\sum_i n_{i\uparrow}n_{i\downarrow} - \mu\sum_{i,\sigma}n_{i\sigma}.$$

Here

$$n_{i\sigma}=c_{i\sigma}^\dagger c_{i\sigma}.$$

Assumptions

• Fermionic lattice modes are used.
• Lattice geometry, boundary conditions, filling, and spin convention are specified.
• $t$ is a hopping amplitude and $U$ is an onsite interaction.
• Longer-range hopping, longer-range interactions, disorder, and phonons are omitted unless added.

Common Mistakes

• Treating the Hubbard model as automatically realistic for every material.
• Forgetting the chemical-potential term when working grand canonically.
• Dropping fermionic signs in finite-basis matrix construction.
• Comparing results without specifying lattice, dimension, filling, and boundary conditions.

Canonical Links

• Hubbard Dimer dossier
• Hubbard Chain dossier
• Hubbard Model
• Hubbard Model
• t–J Model Preview
• Many-Particle Hamiltonians
• Fermionic Anticommutation Relations
• Second-Quantized One-Body Operator

References

• J. Hubbard, “Electron correlations in narrow energy bands”, Proceedings of the Royal Society A 276, 238-257, 1963.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.