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| Anderson model | |
|---|---|
| Name | Anderson model |
| Field | Condensed matter physics |
| Introduced | 1958 |
| Primary contributors | Philip W. Anderson, John Hubbard, P. W. Anderson |
| Related concepts | Localization (physics), Hubbard model, Kondo effect, Fermi liquid theory |
Anderson model The Anderson model is a theoretical framework in condensed matter physics introduced to describe localized electronic states interacting with conducting electrons and impurities; it underpins study of localization (physics), correlated electrons, and magnetism. It connects ideas from Philip W. Anderson's work to techniques developed in John Hubbard's models and has influenced understanding in contexts involving the Kondo problem, Mott insulator, and Fermi liquid theory. The model's versatility makes it central to phenomena explored in research at institutions like Cavendish Laboratory and Bell Labs and in prizes such as the Nobel Prize in Physics.
The Anderson model formulates a localized impurity or orbital hybridized with a conduction band, capturing competition between hybridization, on-site Coulomb repulsion, and disorder; related frameworks include the Hubbard model, the Kondo model, and the s-d model. It informs theoretical work at places including Princeton University, University of Cambridge, Harvard University, and experimental programs at IBM Research. Core topics tied to the model appear in studies by Philip W. Anderson and later developments connected to the Kondo effect resolution by Jun Kondo and renormalization ideas from Kenneth G. Wilson.
Developed in 1958 by Philip W. Anderson to explain localized magnetic moments in metals and the absence of conductivity due to disorder, the model emerged alongside efforts addressing the Kondo problem and the nature of metal–insulator transition. Subsequent contributors included researchers at Bell Labs, theorists such as John Hubbard and Nevill Mott, and methodologists like Kenneth G. Wilson who applied renormalization group ideas. The model influenced interpretations of experiments at facilities like Los Alamos National Laboratory and collaborations involving Columbia University and MIT researchers studying rare-earth compounds and transition-metal oxides.
The canonical single-impurity Anderson Hamiltonian comprises terms for conduction electrons, localized impurity levels, hybridization, and on-site interaction. It is often written in second quantization with operators introduced in texts from Landau Institute-affiliated authors and courses at University of California, Berkeley and École Normale Supérieure. Parameters include the impurity level energy ε_d, hybridization V, conduction band dispersion ε_k, and Coulomb repulsion U; the model links to spectral function analysis used in studies at Max Planck Institute for Solid State Research and formal techniques developed by Abrikosov and Gorkov. Mathematical objects commonly invoked are Green's functions, self-energies, and impurity t-matrix elements derived using approaches from Dyson and Schrieffer–Wolff transformation-style mappings to the Kondo model.
Different parameter regimes produce distinct behaviors: local moment formation akin to observations in heavy-fermion compounds and rare-earth metals, mixed-valence regimes relevant to cerium and uranium intermetallics, and Kondo screening leading to low-temperature Fermi-liquid behavior connected to Wilson's renormalization results. The model captures spectral weight transfer seen in photoemission spectroscopy experiments at Stanford Linear Accelerator Center and signatures in transport measurements examined by teams at Argonne National Laboratory and Brookhaven National Laboratory. It informs understanding of quantum phase transitions studied in contexts like YbRh2Si2 and links to localization physics explored in Anderson localization research.
Exact and approximate methods applied include numerical renormalization group developed by Kenneth G. Wilson, Bethe ansatz solutions for special limits related to work by Nikolai Bogoliubov, quantum Monte Carlo algorithms used in simulations at Oak Ridge National Laboratory, dynamical mean-field theory pioneered by groups at Georges Kotliar's collaborations and implemented at Rutgers University and École Polytechnique, and non-crossing approximation approaches from teams at University of Tokyo. Analytical mappings via the Schrieffer–Wolff transformation relate the Anderson model to the Kondo model and enable perturbative renormalization group analyses by researchers associated with University of Illinois and University of Chicago. Functional renormalization group, slave-boson mean-field theory, and Bethe ansatz variants provide complementary perspectives used in studies at University of Oxford and Columbia University.
Extensions include multi-orbital Anderson models used for transition metal oxides, periodic Anderson lattice models central to heavy-fermion theory and investigations at Los Alamos National Laboratory, and impurity cluster generalizations relevant to high-temperature superconductivity studies at Bell Laboratories and Princeton University. The model underpins computational embedding methods like dynamical mean-field theory applied to materials studied at Argonne National Laboratory's Advanced Photon Source and to nanostructures fabricated at IBM Thomas J. Watson Research Center. It has informed device-level interpretations for quantum dots investigated by groups at Yale University and University of Cambridge and motivated interdisciplinary links to cold-atom emulation efforts at MIT and ETH Zurich.
Empirical signatures of Anderson-type physics appear in spectroscopy of heavy-fermion materials measured at European Synchrotron Radiation Facility and in transport experiments on quantum dot setups at Copenhagen University and University of Geneva. Scanning tunneling microscopy studies at Lawrence Berkeley National Laboratory reveal Kondo resonances consistent with Anderson impurity predictions; neutron scattering experiments at Institut Laue-Langevin probe magnetic fluctuations tied to local moments. Cold-atom simulators at Max Planck Institute for Quantum Optics and JILA have implemented impurity-like Hamiltonians to emulate Anderson physics, while angle-resolved photoemission at SLAC National Accelerator Laboratory and resonant inelastic X-ray scattering at Diamond Light Source detect spectral features predicted by lattice extensions.