Fermi–Hubbard model

Fermi–Hubbard model
Name	Fermi–Hubbard model
Field	Condensed matter physics
Introduced	1963
Authors	John Hubbard; Martin Gutzwiller; Junjiro Kanamori

Fermi–Hubbard model

The Fermi–Hubbard model is a paradigmatic lattice model in condensed matter physics introduced in the early 1960s by John Hubbard, Martin Gutzwiller, and Junjiro Kanamori to describe interacting fermions on a lattice, and it has since become central to studies of high-temperature superconductivity, Mott insulators, and quantum simulation. It provides a minimal Hamiltonian capturing competition between kinetic energy and on-site interaction and has influenced research at institutions such as CERN, MIT, Harvard University, and Max Planck Society while inspiring experiments at facilities like JILA and NIST. The model connects to topics associated with Anderson localization, Heisenberg model, Bardeen–Cooper–Schrieffer theory, and numerical approaches developed at places including Los Alamos National Laboratory and Microsoft Research.

Introduction

The model was formulated by John Hubbard in 1963 to address magnetism in transition metals and was independently analyzed by Martin Gutzwiller and Junjiro Kanamori in the context of electron correlations and itinerant ferromagnetism; it rapidly entered the research programs of groups at Bell Labs, IBM, and the University of Cambridge. The Fermi–Hubbard model sits at the nexus of problems addressed by researchers such as P. W. Anderson, Philip W. Anderson, N. F. Mott, Nigel Goldenfeld, Eugene Wigner, and Lev Landau and links to phenomena studied in experiments at Brookhaven National Laboratory, Argonne National Laboratory, and Rutherford Appleton Laboratory. Its simplicity makes it a benchmark for computational methods developed by teams at Princeton University, Stanford University, Caltech, University of Illinois Urbana–Champaign, and ETH Zurich.

Mathematical formulation

The Hamiltonian is written on a lattice with creation and annihilation operators originally formalized in second quantization by Paul Dirac and used in many-body theory by Richard Feynman and Julian Schwinger; it includes a nearest-neighbor hopping term t and an on-site interaction U, and is often studied on lattices such as square, cubic, honeycomb, and triangular geometries relevant to materials investigated at Bell Labs Research and IBM Research. Analytic and algebraic properties are studied using tools developed by Bethe ansatz proponents like Hans Bethe and algebraic approaches related to work of Elliott Lieb and Freeman Dyson; symmetry considerations trace back to Emmy Noether and group-theoretic methods used by Sophus Lie. The model admits particle-hole transformations exploited in studies by John Bardeen and others, and conserved quantities relate to operators introduced in quantum field theory by Enrico Fermi and Ettore Majorana; numerical discretization schemes draw on techniques developed at Los Alamos National Laboratory and in the Wiener process literature.

Limiting cases and approximate methods

In the weak-coupling limit U << t the model connects to Bardeen–Cooper–Schrieffer theory and perturbative renormalization-group studies by Ken Wilson and Miguel Ángel Virasoro; in the strong-coupling limit U >> t it maps onto the Heisenberg model studied by Werner Heisenberg and leads to superexchange processes described by P. W. Anderson. Exact solutions exist in one dimension via the Bethe ansatz as developed by Hans Bethe and extended by Elliott Lieb and F. Y. Wu; approximate methods include dynamical mean-field theory (DMFT) popularized at Georges Kotliar’s collaborations and quantum Monte Carlo techniques advanced at C. N. Yang’s network and David Ceperley’s group. Variational wavefunctions such as Gutzwiller projection were introduced by Martin Gutzwiller and are used alongside tensor network methods inspired by Steven R. White and matrix product state developments from Guifre Vidal’s school.

Physical properties and phases

The model exhibits a Mott insulating phase described by Nevill Mott at commensurate fillings and antiferromagnetic order connected to the Heisenberg antiferromagnet; doping can produce phases relevant to the phase diagrams of cuprates probed by Zhi-Xun Shen and John Tranquada, and it serves as a theoretical underpinning for proposals of d-wave superconductivity examined by C. C. Tsuei and J. R. Schrieffer. Metallic, superconducting, charge-density-wave, spin-density-wave, and stripe phases have been reported in numerical and experimental studies at Princeton University, Columbia University, and University of California, Berkeley; critical phenomena are analyzed using scaling frameworks developed by Kenneth G. Wilson and conformal field theory methods related to work by Alexander Zamolodchikov. Topological aspects link to research on Quantum Hall effect by Robert Laughlin and to topological orders studied by Xiao-Gang Wen.

Experimental realizations and simulations

Cold-atom experiments in optical lattices realized by groups at MIT, Harvard University, Max Planck Institute of Quantum Optics, and University of Bonn emulate the model with ultracold fermions such as 6Li and 40K; quantum gas microscopy by teams at Max Planck Society and Harvard enables site-resolved measurements. Analog quantum simulations and digital quantum computing implementations have been pursued on platforms from Google’s quantum processors and IBM Quantum hardware to trapped-ion systems developed at NIST and IonQ, while arrays of quantum dots studied at Sandia National Laboratories and LBNL offer solid-state emulation. Neutron scattering experiments at ISIS Neutron and Muon Source and synchrotron studies at Diamond Light Source probe correlation effects in materials approximated by the model, and ultrafast spectroscopy groups at SLAC National Accelerator Laboratory investigate nonequilibrium dynamics.

Extensions and related models

Extensions include the Hubbard–Holstein model coupling electrons to phonons studied by Holstein and collaborators, multi-orbital Hubbard models relevant to iron pnictides investigated at Rice University, and Kondo–Hubbard hybrids related to the Kondo effect explored by Jun Kondo; the t–J model derived in the large-U limit was used by Patrick Lee and co-workers to study high-Tc superconductivity. Related lattice models include the Anderson impurity model by Philip W. Anderson, the Falicov–Kimball model introduced by L. M. Falicov and J. C. Kimball, and quantum spin models such as the XXZ model and Kitaev model studied by Alexei Kitaev; modern developments tie the Fermi–Hubbard framework to tensor network approaches from Roman Orús and quantum embedding methods developed by groups at ETH Zurich and Rutgers University.

Category:Condensed matter physics