| Hubbard model | |
|---|---|
| Name | Hubbard model |
| Caption | Schematic of electrons hopping on a lattice with on-site interaction U |
| Introduced | 1963 |
| Creators | John Hubbard |
| Field | Condensed matter physics |
| Related | t–J model, Anderson impurity model, Heisenberg model |
Hubbard model
The Hubbard model is a minimal quantum many-body model describing interacting fermions on a lattice, capturing the competition between kinetic energy from hopping and local interaction energy. It plays a central role in Condensed matter physics and Quantum many-body theory as a paradigmatic description of magnetism, metal–insulator transitions, and unconventional superconductivity. Because of its simplicity and rich behavior, the model informs both theoretical advances and experiments in correlated materials and ultracold atoms.
The Hubbard model was introduced by John Hubbard in 1963 to study electron correlation effects in narrow-band solids. It encapsulates two essential processes: nearest-neighbor hopping (parameter t) that delocalizes electrons, and on-site Coulomb repulsion (parameter U) that penalizes double occupancy. Physically it addresses problems such as the Mott insulator transition first discussed by Nevill Mott and provides a lattice-level complement to continuum approaches like the Kohn–Sham DFT and the Hartree–Fock method. The model is defined on various lattice geometries—including the square lattice, triangular lattice, and Bethe lattice—and is a cornerstone for understanding strongly correlated electron systems such as cuprate superconductors, transition metal oxides, and organic conductors.
The standard Hubbard Hamiltonian 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} creates a fermion with spin σ on site i, n_{iσ}=c^†_{iσ}c_{iσ}, and μ is the chemical potential. Variants include the multi-orbital Hubbard model used for transition metal compounds, the extended Hubbard model with nearest-neighbor interaction V, the attractive Hubbard model (U<0) relevant for s-wave pairing, and the Hubbard–Holstein model coupling electrons to phonons. Strong-coupling expansions connect the Hubbard model to the Heisenberg model (via superexchange J≈4t^2/U) and to the t–J model in the large-U limit. The model can be defined for fermions or mapped to bosonic analogues (Bose–Hubbard model) studied in optical lattice experiments.
Exact solutions exist only in special cases, notably the one-dimensional Hubbard chain solved by the Bethe ansatz (Lieb–Wu solution). For higher dimensions or realistic multi-orbital problems, a variety of analytical and numerical methods are employed: perturbation theory, mean-field approximations (e.g., Hartree–Fock, DMFT), quantum Monte Carlo (QMC), density matrix renormalization group (DMRG), tensor network states, and variational Monte Carlo. DMFT, developed by Georges, Kotliar, Krauth and others, maps the lattice problem to an effective Anderson impurity model solved self-consistently and has been combined with DFT+DMFT for materials modeling. Advances in quantum computing and hybrid algorithms (e.g., variational quantum eigensolver) aim to tackle Hubbard-like Hamiltonians on devices by groups at IBM, Google, and academic centers such as Harvard University and MIT.
The Hubbard model exhibits a rich phase diagram: metallic, Mott insulating, antiferromagnetic, charge-ordered, and superconducting phases appear depending on lattice, density, U/t, and temperature. On the half-filled square lattice, antiferromagnetism emerges at strong coupling; on doping, models of high-temperature superconductivity—particularly d-wave pairing—have been proposed and intensely studied in the context of cuprate superconductors and proposals by P. W. Anderson's resonating valence bond (RVB) theory. Quantum critical points and non-Fermi-liquid behavior are predicted near phase boundaries. Topological and frustrated lattice variants produce spin-liquid states studied in connection with quantum spin liquids and materials like herbertsmithite.
Real-material realizations include transition metal oxides, nickelates, and cuprates, where correlations approximated by Hubbard physics drive insulating and magnetic behavior. Ultracold atoms in optical lattices provide a clean, tunable platform to realize the Hubbard Hamiltonian; landmark experiments by groups at MIT, JILA, Harvard, and Max Planck Institute for Quantum Optics have simulated the fermionic Hubbard model, observed Mott insulating states, and probed short-range correlations. Quantum gas microscopy enables site-resolved observation of Hubbard dynamics and entanglement. Solid-state quantum simulators and engineered materials (e.g., moiré heterostructures like twisted bilayer graphene) extend Hubbard-like descriptions to designer correlated systems.
The Hubbard model serves as a testing ground for theoretical concepts: emergent quasiparticles, collective modes, renormalization group flows, and entanglement structure in many-body states. It motivates development of methods in quantum Monte Carlo, tensor networks, and dynamical mean field theory, and intersects with fields like quantum information when studying entanglement entropy and simulation complexity. The model links microscopic Hamiltonians to phenomenology in materials science and informs computational materials discovery efforts at institutions such as Argonne National Laboratory and Lawrence Berkeley National Laboratory.
Understanding Hubbard physics is central to designing correlated-electron materials with targeted properties—superconductors, high-capacity battery cathodes, or quantum materials for information technologies—which has economic and societal implications for energy and computing. Open problems include definitive resolution of superconductivity mechanisms in doped two-dimensional models, controlled treatment of non-equilibrium dynamics, and scalability of quantum simulations. Questions of equitable access to emerging quantum technologies and responsible material deployment foreground ethical considerations; research communities at universities, national labs, and industry collaborate to ensure inclusive benefits from advances grounded in Hubbard-model physics.
Category:Condensed matter physics Category:Quantum many-body theory