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| Landau–Ginzburg model | |
|---|---|
| Name | Landau–Ginzburg model |
| Field | Theoretical physics |
| Introduced | 1950s |
| Contributors | Lev Landau, Vitaly Ginzburg |
| Related | Ginzburg–Landau theory, Ginzburg–Landau equations, Bardeen–Cooper–Schrieffer theory |
Landau–Ginzburg model The Landau–Ginzburg model is a phenomenological field theory introduced in the 1950s to describe phase transitions and order parameters in condensed matter physics. It bridges microscopic frameworks such as Bardeen–Cooper–Schrieffer theory and macroscopic descriptions like the Ginzburg–Landau equations while informing developments in quantum field theory, statistical mechanics, and string theory. The model has influenced research at institutions including Moscow State University, Princeton University, and CERN, and figures in work by Lev Landau, Vitaly Ginzburg, Lev Pitaevskii, and Philip W. Anderson.
The Landau–Ginzburg model formulates a free-energy functional for an order parameter field, connecting to ideas in Landau theory of phase transitions, mean field theory, and critical phenomena. It provides a continuum description applicable near critical points studied by researchers at University of Cambridge, Harvard University, and École Normale Supérieure, and underpins methods used in analyses by Kenneth Wilson, Michael Fisher, and Leo Kadanoff. The model serves as a bridge between experimental results from laboratories like Bell Labs and theoretical constructs developed at Institute for Advanced Study.
The origin lies in the collaboration between Lev Landau and Vitaly Ginzburg in the late 1940s and early 1950s, building on insights from Landau theory of phase transitions and experimental puzzles addressed at Kapitza Institute for Physical Problems. Its rise paralleled breakthroughs such as Bardeen–Cooper–Schrieffer theory for superconductivity and influenced later work by Abrikosov on vortices and by Anderson on broken symmetry. The model became central to the theoretical programs of groups at Bell Labs, Los Alamos National Laboratory, and Moscow Institute of Physics and Technology, and featured in conferences like those at Solvay Conference and International Congress of Mathematicians where links to renormalization group and critical exponents were explored by Kenneth Wilson and Michael Fisher.
The canonical Landau–Ginzburg functional expresses free energy as an integral of a local Lagrangian density depending on an order parameter field φ(x) and its gradients, mirroring constructions in Euclidean quantum field theory and echoing variational principles used at Princeton University and University of Chicago. The action-like expression includes mass terms, quartic interactions, and gradient terms reminiscent of those in models studied by Julian Schwinger, Richard Feynman, and Gerard 't Hooft. Functional integrals and saddle-point approximations invoke techniques developed by Paul Dirac, John von Neumann, and Emmy Noether; symmetry-breaking patterns connect to classification schemes influenced by Élie Cartan and Hermann Weyl.
Applied to superconductivity, the model connects to Bardeen–Cooper–Schrieffer theory and to vortex solutions first studied by Alexei Abrikosov; in magnetism it describes phenomena investigated at Los Alamos National Laboratory and Argonne National Laboratory. In liquid crystals and superfluidity the framework complements experiments at Max Planck Society and Cavendish Laboratory, interfacing with theories by Lev Pitaevskii and Evgeny Lifshitz. Cosmological analogues informed work at Caltech and Yale University on defect formation discussed by Tom Kibble and Andrei Linde, while applications in low-dimensional systems relate to studies by J. Michael Kosterlitz and David Thouless.
Extensions include supersymmetric Landau–Ginzburg models developed in the context of supersymmetry and explored by groups at Stanford University, Rutgers University, and Santa Barbara. Gauged versions couple the order parameter to gauge fields as in studies by Yoichiro Nambu and Gerard 't Hooft, while stochastic variants connect to non-equilibrium work at Los Alamos National Laboratory and Imperial College London. Lattice discretizations tie to numerical programs at Lawrence Berkeley National Laboratory and National Institute for Theoretical Physics, and multi-component generalizations link to classification efforts by Philip W. Anderson and C. N. Yang.
Certain low-dimensional and integrable instances admit exact treatment, as pursued in collaborations across Institute for Advanced Study and Perimeter Institute for Theoretical Physics. Two-dimensional minimal models and soluble potentials relate to techniques advanced by Alexander Zamolodchikov, Al. B. Zamolodchikov, and Vladimir Fateev; exact renormalization analyses draw on methods developed by Kenneth Wilson and Michael Fisher. Solvable lattice realizations connect to results by Rodney Baxter and to conformal bootstrap studies influenced by Alexander Polyakov.
In two dimensions, Landau–Ginzburg models flow under renormalization to fixed points described by conformal field theory, a correspondence elucidated in work by Edward Witten, Philip Candelas, and Cumrun Vafa. This relationship underlies homological mirror symmetry programs at Institute for Advanced Study and Harvard University, and informs string compactification studies by Gary Horowitz and Shamit Kachru. Mirror symmetry connections were developed in collaborations involving Maxim Kontsevich and Paul Seidel, linking Landau–Ginzburg potentials to categories studied at MPI for Mathematics and Mathematical Institute, Oxford.