| Hubbard model | |
|---|---|
| Name | Hubbard model |
| Caption | Schematic of electrons on a lattice with hopping t and on-site interaction U |
| Type | Theoretical model |
| Field | Condensed matter physics |
| Introduced | 1963 |
| Authors | John Hubbard |
| Related | Anderson impurity model, t–J model, Heisenberg model |
Hubbard model
The Hubbard model is a foundational lattice model in Condensed matter physics and Quantum Physics that describes interacting fermions on a lattice via nearest-neighbour hopping and an on-site interaction. It provides a minimal framework to study electron correlation phenomena such as magnetism, metal–insulator transitions and superconductivity, and is central to modern approaches to strongly correlated materials and quantum simulation.
The Hubbard model was introduced by John Hubbard (and independently related work by Martin Gutzwiller and Junjiro Kanamori) to capture essential physics of electrons in narrow-band solids where the competition between kinetic energy and Coulomb repulsion is dominant. It sits between single-particle band theory (e.g. Bloch theorem, Fermi surface) and full quantum electrodynamical treatments, emphasizing low-energy lattice degrees of freedom. The model is invoked to explain phenomena in transition-metal oxides such as cuprate superconductors, manganites, and nickelates, and informs modern experiments with ultracold atoms in optical lattice potentials and engineered quantum simulator platforms provided by groups at institutions like MIT, Harvard University, Max Planck Society, and ETH Zurich.
The Hubbard Hamiltonian on a lattice Λ is typically written as H = - t ∑_{⟨i,j⟩,σ} (c^†_{iσ} c_{jσ} + h.c.) + U ∑_i n_{i↑} n_{i↓} - μ ∑_{i,σ} n_{iσ}, where c^†_{iσ} (c_{iσ}) create (annihilate) a fermion with spin σ on site i, n_{iσ}=c^†_{iσ}c_{iσ}, t is the hopping amplitude and U the on-site repulsion. Variants include extended Hubbard models with nearest-neighbour interaction V, multi-orbital Hubbard models important for transition metal physics, and Hubbard–Holstein couplings to lattice phonon modes. The model connects formally to the Anderson impurity model via dynamical mean-field mappings and to the Heisenberg model in the strong-coupling limit.
Several limits of the Hubbard model admit exact or controlled solutions. The noninteracting limit U=0 reduces to tight-binding band theory solvable by Fourier transform. The atomic limit t=0 is trivial and reveals Hubbard bands. The one-dimensional Hubbard model is solvable by the Bethe ansatz (Lieb and Wu solution), giving exact charge and spin excitation spectra and demonstrating spin–charge separation. At half-filling and large U/t the model maps to the Heisenberg model with exchange J ≈ 4t^2/U. Certain finite clusters and bipartite lattices have rigorous results (e.g. Nagaoka ferromagnetism for a single hole in the U→∞ limit on particular lattices). Integrable and asymptotic analyses inform understanding of metal–insulator transitions (the Mott transition).
A wide toolbox addresses the Hubbard problem. Analytical techniques include perturbation theory, strong-coupling expansions, mean-field theories such as Hartree–Fock, slave-boson and slave-fermion methods, variational wavefunctions (e.g. Gutzwiller approximation), and field-theory approaches (bosonization in 1D, renormalization group). Numerical methods include exact diagonalization, Quantum Monte Carlo (with sign-problem considerations), density matrix renormalization group (DMRG) for 1D systems, tensor network states (e.g. matrix product states, projected entangled pair states), and dynamical mean-field theory (DMFT) and cluster extensions (CDMFT, DCA). Modern developments couple these to high-performance computing at centers like Oak Ridge National Laboratory and CERN collaborations and to quantum computing experiments from IBM Quantum and Google Quantum AI.
The Hubbard model exhibits diverse collective phases depending on lattice geometry, band filling and interaction strength: metallic Fermi-liquid behaviour, antiferromagnetism at half-filling on bipartite lattices, ferromagnetism in special limits, charge density waves, and unconventional superconductivity (notably proposed d-wave pairing in the two-dimensional square-lattice Hubbard model relevant to cuprates). It captures the formation of Hubbard bands and the Mott insulating state, spin–charge separation in 1D, and pseudogap phenomena observed in correlated materials. Competing orders and quantum criticality are studied using finite-temperature phase diagrams and spectral functions derived from DMFT, cluster methods, and angle-resolved photoemission spectroscopy (ARPES) comparisons.
Real materials approximating the Hubbard model include transition metal oxides, layered cuprates, organic conductors, and certain heavy fermion systems. Experiments probe Hubbard physics via transport, neutron scattering, ARPES, scanning tunnelling microscopy (STM), and optical spectroscopy. Ultracold fermionic atoms (e.g. ^6Li, ^40K) in optical lattices have realized tunable Hubbard Hamiltonians allowing direct measurement of correlation functions, observation of the Mott plateau, and studies of spin correlations — performed in groups at MIT, University of Cambridge, Institute of Quantum Optics and Quantum Information, and elsewhere. Artificial materials such as moiré heterostructures (e.g. twisted bilayer graphene) display effective Hubbard-like behavior, linking model predictions to device engineering.
The Hubbard model is a pedagogical cornerstone in courses on many-body physics, solid state physics, and computational quantum physics, providing a clear, conservative framework to teach electron correlation and emergent collective behaviour. It motivates advanced topics such as correlated electron methods, quantum phase transitions, and modern numerical techniques. In research it remains a unifying paradigm guiding exploration of emergent phenomena in complex materials, quantum simulators, and potential routes to engineered superconductivity and quantum technologies, reinforcing institutional efforts across universities and national laboratories to preserve and extend core knowledge in condensed matter and quantum science.
Category:Condensed matter physics Category:Quantum many-body theory Category:Lattice models