Ising model in nLab (original) (raw)
Contents
Context
Algebraic Quantum Field Theory
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
Concepts
quantum mechanical system, quantum probability
- subsystem
- observables
- local net of observables
- state on a star-algebra, expectation value
- picture of quantum mechanics
- gauge symmetry
- BRST complex, BV-BRST formalism
- local BV-BRST complex
- BV-operator
- quantum master equation
- master Ward identity
- gauge anomaly
interacting field quantization
- causal perturbation theory, perturbative AQFT
- interaction
- S-matrix, scattering amplitude
- interacting field algebra
- adiabatic limit
Theorems
States and observables
- order-theoretic structure in quantum mechanics
- Kochen-Specker theorem
- Bell's theorem
- Fell's theorem
- Gleason's theorem
- Wigner theorem
- Bub-Clifton theorem
- Kadison-Singer problem
Operator algebra
- Wick's theorem
- GNS construction
- modular theory
- Fell's theorem
- Stone-von Neumann theorem
- Haag's theorem
Local QFT
- Reeh-Schlieder theorem
- Bisognano-Wichmann theorem
- PCT theorem
- spin-statistics theorem
- DHR superselection theory
- Osterwalder-Schrader theorem (Wick rotation)
Perturbative QFT
Functorial Quantum field theory
functorial quantum field theory
Contents
- cobordism category
- cobordism hypothesis
- functorial field theory
- FQFT and cohomology
- holographic principle of higher category theory
Contents
Idea
The Ising model is a simple lattice model (in theoretical physics) of physical systems roughly similar to ferromagnets. Its configurations are functions on a lattice with values in {−1,+1}\{-1,+1\}, roughly to be thought of as the magnetic polarizations of elementary magnets in a crystal lattice.
The Ising model gained importance as a toy model in theoretical physics. It is about the simplest possible model that allows methods of Euclidean quantum field theory in statistical physics and the study of critical phenomena?. In fact at a critical temperature and in dimension 2 the model exibits the behaviour of a 2dconformal field theory, one of the examples of rational conformal field theory.
Real-world physical systems that show behaviour described by the Ising model were only found later (Wolf).
Properties
References
On historical work by Yoichiro Nambu:
- Lars Brink, Pierre Ramond, Nambu and the Ising Model [arXiv:2209.01122]
Original articles:
- Franz Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters, Journal of Mathematical Physics 12, 2259 (1971) (doi:10.1063/1.1665530)
(introducing the idea of lattice QCD)
(…)
Review:
- Kasper Peeters, Marija Zamaklar, section 2.1 of Euclidean Field Theory, Lecture notes 2009-2011 (web, pdf)
- W. P. Wolf, The Ising model and real magnetic materials, Braz. J. Phys. vol.30 no.4 São Paulo Dec. 2000 (web)
See also
- Wikipedia, Ising model
Discussion of the Ising model 2d CFT as a boundary theory to a 3d TQFT based on the Turaev-Viro model, similar to the CS-WZW correspondence, and the phenomenon of Kramers-Wannier duality, is in:
- Jürg Fröhlich, Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Kramers-Wannier duality from conformal defects (arXiv:cond-mat/0404051)
(via FRS-formalism) and also in
- Daniel Freed, Constantin Teleman, Topological dualities in the Ising model (arXiv:1806.00008)
3d version:
- Connor Behan, Bootstrapping the long-range Ising model in three dimensions, Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 7 (arXiv:1810.07199, doi:10.1088/1751-8121/aafd1b)
Last revised on September 6, 2022 at 07:08:31. See the history of this page for a list of all contributions to it.