Hubbard Model

One-Sentence Description

The Hubbard model is a minimal lattice model in which fermions hop between sites and interact when opposite-spin particles occupy the same site.

Physical Setup

Choose a lattice, place fermionic modes on each site and spin label, and allow nearest-neighbor hopping plus onsite repulsion or attraction. The model is a controlled abstraction for correlated electrons, optical lattices, Mott physics, magnetism, and finite-cluster benchmarks. The Hubbard Dimer dossier gives the convention-complete two-site record, while the Hubbard Chain dossier records the one-dimensional Lieb–Wu solution and four-site benchmark.

Hilbert Space

For $L$ sites and two spin labels, there are $2L$ fermionic modes. The full fermionic Fock space has dimension

$$\dim\mathcal F=2^{2L}.$$

Fixed-particle-number and fixed-spin sectors are often used to reduce calculations.

Hamiltonian

A standard spinful fermionic form 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

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

Parameters

Symbol	Meaning
$t$	hopping amplitude
$U$	onsite interaction energy
$\mu$	chemical potential
$L$	number of sites
$n$	filling or average particles per site

Solvability

The Hubbard model is not generally exactly solvable. Special cases are controlled: $U=0$ gives a free lattice Fermi gas; $t=0$ gives decoupled sites; one-dimensional versions have Bethe-ansatz solvability in important cases; small clusters can be exactly diagonalized.

At large positive $U$ near half filling, low-energy spin physics can reduce to an antiferromagnetic exchange model with scale

$$J_{\mathrm{eff}} \sim \frac{4t^2}{U}.$$

Key Observables

• Filling and double occupancy.
• Spin and charge correlations.
• Spectral function and density of states.
• Mott gap diagnostics.
• Pairing correlations for attractive or doped variants.

What It Teaches

The Hubbard model teaches how a simple local interaction can produce strong correlation physics beyond band theory. It is the canonical bridge from second-quantized lattice Hamiltonians to Mott insulators, magnetism, and numerical many-body methods.

Canonical Links

• Hubbard Dimer dossier
• Hubbard Chain dossier
• Hubbard Model
• Hubbard Model Glossary Entry
• Hubbard Model Hamiltonian
• t–J Model Preview
• Many-Particle Hamiltonians
• Fermionic Anticommutation Relations
• Fock Space Examples

Variants

• repulsive Hubbard model;
• attractive Hubbard model;
• single-band and multi-band Hubbard models;
• extended Hubbard models with longer-range interactions;
• Hubbard ladders and clusters;
• continuum-to-lattice Hubbard approximations for optical lattices.

Common Mistakes

• Treating the Hubbard model as automatically realistic for every material.
• Forgetting to specify lattice geometry and filling.
• Dropping fermionic signs in basis construction.
• Comparing $U/t$ across conventions without checking the hopping normalization.

Quick Check

Why is the onsite interaction $Un_{i\uparrow}n_{i\downarrow}$ a two-body term even though it is written at one site?

Solution

It contributes only when two fermions with opposite spin occupy the same site. The factor is a product of two number operators, so it counts an onsite pair, not a one-particle energy.

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.