| Hubbard model | |
|---|---|
| Name | Hubbard model |
| Introduced | 1963 |
| Founder | John Hubbard |
| Field | Condensed matter physics |
| Applications | High-temperature superconductivity, magnetism, Ultracold atomic gases |
Hubbard model
The Hubbard model is a simple lattice model in condensed matter physics that describes interacting electrons hopping between sites with an on-site repulsion. It is a paradigmatic model for studying correlation-driven phenomena such as Mott insulating behavior, magnetic order, and superconductivity in strongly correlated materials. Introduced by John Hubbard in 1963, it remains central to analytical, numerical and experimental studies bridging models such as the Anderson impurity model and techniques like dynamical mean field theory.
The Hubbard model was proposed to capture the competition between the kinetic energy of itinerant electrons and the local Coulomb repulsion that favors localization. In many transition metal oxides and organic conductors, narrow bands and significant on-site repulsion produce phenomena unexplained by independent-electron theories such as band theory. The model provides a minimal framework to explore the emergence of Mott metal–insulator transitions, antiferromagnetism, and unconventional pairing mechanisms implicated in cuprate superconductors and other correlated systems studied at institutions such as Bell Labs and Cavendish Laboratory. Its conceptual proximity to the Heisenberg model and the t–J model makes it a unifying platform in theoretical solid state physics.
The canonical single-band Hubbard Hamiltonian on a lattice Λ is 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}) creates (annihilates) an electron of spin σ at site i, t is the nearest-neighbor hopping amplitude, U the on-site interaction, and μ the chemical potential. The lattice may be a square lattice, cubic lattice, triangular lattice or other graph; special cases include one-dimensional chains where exact methods apply. At large U/t the model maps perturbatively onto the Heisenberg model with exchange J ≈ 4t^2/U, while at weak coupling it connects to Fermi liquid theory and weak-coupling instabilities treated by BCS theory or random phase approximation.
No general closed-form solution exists for the Hubbard model in dimensions greater than one, so a variety of analytical and numerical methods are used: - Exact solutions: the one-dimensional Hubbard model was solved by the Lieb–Wu solution using the Bethe ansatz. - Field-theory and perturbative methods: bosonization and renormalization group analyses characterize low-energy behavior in one dimension; diagrammatic expansions (e.g., GW approximation) address weak coupling. - Numerical many-body techniques: exact diagonalization, quantum Monte Carlo (with sign problem issues), density matrix renormalization group (DMRG) effective in 1D and quasi-1D, and tensor network methods (e.g., matrix product states and projected entangled pair states) for higher dimensions. - Dynamical mean field theory (DMFT) and cluster extensions (CDMFT, DCA) map the lattice problem to a quantum impurity solved by impurity solvers such as numerical renormalization group or continuous-time quantum Monte Carlo; DMFT captures local correlations and the Mott transition. - Variational approaches: Gutzwiller and Jastrow wavefunctions, variational Monte Carlo, and slave-particle formalisms (slave boson, slave rotor) yield approximate phase diagrams and quasiparticle renormalizations.
The Hubbard model exhibits a rich phase diagram dependent on lattice geometry, band filling, temperature, and U/t: - At half-filling on bipartite lattices and large U, the ground state is an antiferromagnetic Mott insulator with low-energy spin dynamics described by the Heisenberg model. - Doping away from half-filling can produce metallic states, possible d-wave superconductivity particularly on the square lattice relevant to cuprates, and stripe or charge-density-wave order. - The model captures the Mott metal–insulator transition, including quasiparticle weight collapse and Hubbard bands visible in spectral functions, as elucidated by DMFT and photoemission experiments. - Frustrated lattices (e.g., triangular lattice) and multi-band generalizations can stabilize nontrivial phases such as quantum spin liquids, charge order, or ferromagnetism via the Nagaoka mechanism in extreme limits.
Numerous extensions connect the Hubbard model to broader correlated-electron physics: - Multi-orbital Hubbard models incorporate Hund's coupling and orbital degrees of freedom, relevant to iron-based superconductors and transition metal oxides. - The Hubbard–Holstein model couples electrons to phonons and captures polaronic effects and electron–phonon interplay. - The t–J model arises as a large-U effective theory emphasizing superexchange and constrained Hilbert space, used extensively in studies of high-Tc superconductivity. - The periodic Anderson model and Kondo lattice model generalize localized–itinerant interplay relevant to heavy fermion materials and exotic quantum criticality investigated at institutions like Los Alamos National Laboratory and Max Planck Institute for Solid State Research. - Long-range interactions and spin–orbit coupling produce further variants applied to topological phases and correlated iridates.
Real materials approximated by the Hubbard model include transition metal oxides (e.g., La2CuO4 parent compounds of cuprates), organic charge-transfer salts, and alkali-doped fullerides. Experimental probes such as angle-resolved photoemission spectroscopy (ARPES), optical conductivity, and neutron scattering have observed Hubbard bands, spin excitations, and Mott gaps consistent with model predictions. Ultracold atoms in optical lattices provide a clean, tunable platform for realizing the Hubbard Hamiltonian with control over t, U, and lattice geometry; landmark experiments at MIT and Harvard University have measured the Mott transition and antiferromagnetic correlations. Quantum simulation efforts using quantum gas microscopes and analog quantum simulators continue to test Hubbard physics and its extensions.
Category:Condensed matter physics Category:Models in physics