Hubbard model

Hubbard model
Susumu Yamada, Toshiyuki Imamura, Masahiko Machida · CC BY-SA 4.0 · source
Name	Hubbard model
Field	Condensed matter physics
Introduced	1963
Creators	John Hubbard
Notable for	Strongly correlated electrons

Hubbard model The Hubbard model is a minimal theoretical model for interacting electrons in a lattice introduced in 1963 by John Hubbard, designed to capture competition between electron kinetic energy and on-site Coulomb repulsion. It underpins quantitative and qualitative studies of magnetism, metal–insulator transitions, superconductivity, and quantum phase transitions in systems ranging from transition metal oxides to ultracold atoms. The model is central to research programs at institutions such as Cavendish Laboratory, Bohr Institute, Max Planck Institute for Solid State Research, and has deeply influenced work by figures including Philip W. Anderson, Walter Kohn, Nevill Mott, Kenneth G. Wilson, and Alexei Abrikosov.

Introduction

The Hubbard model was proposed to explain phenomena associated with narrow-band materials such as the Mott insulator behavior observed in early studies of transition metal oxides like NiO and V2O3. It became a cornerstone in the development of theories of high-temperature superconductivity following experiments on CuO2 planes in cuprate materials such as La2CuO4 and theoretical proposals by Philip W. Anderson. The model is often compared and contrasted with the Heisenberg model, the t-J model, and the Anderson impurity model in studies conducted at centers including Bell Labs and Los Alamos National Laboratory.

Mathematical formulation

The canonical Hubbard Hamiltonian on a lattice Λ is written in second quantized form with creation and annihilation operators that obey Fermi statistics, combining a nearest-neighbor hopping term t and an on-site interaction U. In addition to the original single-band formulation used to model compounds like BaBiO3, multi-band and orbital-generalized Hamiltonians incorporate Hund’s coupling terms relevant to materials studied at Oak Ridge National Laboratory. Lattice geometries frequently considered include the square lattice, triangular lattice, honeycomb lattice, and fcc or bcc lattices relevant to elemental metals; boundary conditions such as periodic or open are applied in numerical studies by groups at Argonne National Laboratory.

Methods of solution

Exact solutions exist in special cases, notably the one-dimensional chain solved via the Bethe ansatz by researchers building on techniques from Lieb–Liniger model studies, while higher-dimensional cases require a variety of analytical and numerical approaches. Analytical methods include perturbation theory related to Feynman diagram expansions, strong-coupling expansions linked to the t-J model derivation, slave-boson and slave-fermion mean-field theories developed in the tradition of work at Princeton University, and renormalization group methods pioneered by Kenneth G. Wilson. Numerical and computational techniques central to Hubbard-model research include exact diagonalization used in studies at IBM Research, quantum Monte Carlo methods developed to study the sign problem, density matrix renormalization group (DMRG) originating from Ian Affleck and Steven R. White work, cluster dynamical mean-field theory (CDMFT) and dynamical cluster approximation (DCA) extensions of Dynamical mean-field theory (DMFT) advanced by researchers at Rutgers University and ETH Zurich, and tensor network methods applied by teams at Caltech and MPI für Physik komplexer Systeme.

Phases and physical properties

The Hubbard model exhibits a rich phase diagram including antiferromagnetism, ferromagnetism, Mott insulating phases, metallic phases, and unconventional superconductivity; phase boundaries have been mapped in studies by collaborations involving Bell Labs, MIT, and Stanford University. At half-filling on bipartite lattices the strong-coupling limit maps to the Heisenberg model and produces Néel order as observed in neutron scattering experiments at facilities like Oak Ridge National Laboratory's Spallation Neutron Source. Doped regimes yield competition between stripe order studied in La2-xSrxCuO4 experiments and d-wave superconductivity theorized for cuprates such as YBa2Cu3O7; theoretical predictions have informed spectroscopic measurements at SLAC National Accelerator Laboratory and Brookhaven National Laboratory beamlines. Quantum criticality and pseudogap behavior in two dimensions have been central topics for researchers at Columbia University and Harvard University.

Extensions and variants

Extensions include the multi-orbital Hubbard model incorporating Hund’s coupling relevant to iron-based superconductors like FeSe and FeAs compounds, the ionic Hubbard model applied to charge-transfer salts such as BEDT-TTF salts investigated at Max Planck Institute for Chemical Physics of Solids, and the Hubbard–Holstein model coupling electrons to phonons studied in molecular solids such as K3C60. Long-range Coulomb extensions, spin–orbit coupled variants relevant to Sr2IrO4, and disordered Hubbard models address phenomena in doped semiconductors and amorphous systems explored at Sandia National Laboratories. Effective low-energy models like the t-J model and Kondo lattice models emerge from Schrieffer–Wolff transformations in strong-coupling analyses performed by teams at Los Alamos National Laboratory.

Applications and experimental realizations

Realizations and applications span correlated electron materials, cold atom quantum simulators, and engineered heterostructures. Ultracold fermionic atoms in optical lattices created at laboratories such as MIT and University of Chicago implement Hubbard Hamiltonians with tunable t and U using Feshbach resonances and lattice depth control; these platforms have demonstrated Mott transitions and short-range magnetic correlations. Layered transition metal oxides and heterostructures fabricated at Bell Labs and IBM provide solid-state realizations where angle-resolved photoemission spectroscopy (ARPES) at facilities like SLAC National Accelerator Laboratory probes predicted band renormalizations. The model also informs understanding of metal–insulator transitions in compounds studied at Argonne National Laboratory and guides materials design in projects at National Renewable Energy Laboratory.

Category:Condensed matter physics