"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| − | |||
* roles in the following fields | * roles in the following fields | ||
| − | ** | + | ** [[Integrable systems and solvable models]] |
| − | * | + | * exact solvability of many models is explained by commuting transfer matrices |
| − | ** quantum groups | + | ** [[quantum groups]] |
| − | ** | + | ** [[Conformal field theory (CFT)]] |
| − | ** | + | ** [[Topological quantum field theory(TQFT)]] |
** braid groups | ** braid groups | ||
| − | + | * manifestations of Yang-Baxter equation | |
| − | * manifestations of Yang-Baxter equation | + | ** [[Exact S-matrices in ATFT]] |
| − | ** | + | * <math>R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}</math> |
| − | * <math>R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}</math | + | * for vertex models, YBE becomes the star-triangle relation |
| − | * for vertex models, YBE becomes the star-triangle relation | + | * see '''[Baxter1995] '''for a historical account |
| − | * see '''[Baxter1995] '''for a historical account | + | |
| + | |||
==Yang and Baxter== | ==Yang and Baxter== | ||
| − | * '''[Yang1967]''' [[interacting particles with potential]] | + | * '''[Yang1967]''' [[interacting particles with potential]] |
| − | ** Bethe ansatz gave rise to an equation | + | ** Bethe ansatz gave rise to an equation |
| − | * '''[Baxter1972] '''considered the problem of [[eight-vertex model and quantum XYZ model]] | + | * '''[Baxter1972] '''considered the problem of [[eight-vertex model and quantum XYZ model]] |
| − | ** commutation of transfer matrices | + | ** commutation of transfer matrices |
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==Bethe ansatz== | ==Bethe ansatz== | ||
| − | * [[Bethe ansatz]] amplitude | + | * [[Bethe ansatz]] amplitude |
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==integrability of a model== | ==integrability of a model== | ||
| − | * in the space of couplings a submanifold exists, such as that the transfer matrices corresponding to any two points P and P' on it commute | + | * in the space of couplings a submanifold exists, such as that the transfer matrices corresponding to any two points P and P' on it commute |
| − | * characterized by a set of equations on the Boltzmann weights | + | * characterized by a set of equations on the Boltzmann weights |
| − | ** this set of equations is called the Yang-Baxter equation | + | ** this set of equations is called the Yang-Baxter equation |
| − | * solutions to Yang-Baxter equation can lead to a construction of integrable models | + | * solutions to Yang-Baxter equation can lead to a construction of integrable models |
| 47번째 줄: | 47번째 줄: | ||
==transfer matrix== | ==transfer matrix== | ||
| − | * borrowed from [[transfer matrix in statistical mechanics]] | + | * borrowed from [[transfer matrix in statistical mechanics]] |
| − | * transfer matrix is builtup from matrices of Boltzmann weights | + | * transfer matrix is builtup from matrices of Boltzmann weights |
| − | * we need the transfer matrices coming from different set of Boltzman weights commute | + | * we need the transfer matrices coming from different set of Boltzman weights commute |
| − | * partition function = trace of power of transfer matrices | + | * partition function = trace of power of transfer matrices |
| − | * so the problem of solving the model is reduced to the computation of this trace | + | * so the problem of solving the model is reduced to the computation of this trace |
| 59번째 줄: | 59번째 줄: | ||
==R-matrix== | ==R-matrix== | ||
| − | * we make a matrix from the Boltzmann weights | + | * we make a matrix from the Boltzmann weights |
| − | * if we can find an R-matrix, then it implies the existence of a set of Boltzmann weights which give exactly solvable models | + | * if we can find an R-matrix, then it implies the existence of a set of Boltzmann weights which give exactly solvable models |
| − | * that is why we care about the quantum groups | + | * that is why we care about the quantum groups |
| − | * spectral parameters | + | * spectral parameters |
| − | * anistropy parameters | + | * anistropy parameters |
| − | * with an R-matrix satisfying the YBE, we obtain a representation of the [[Braid group]], which then gives a link invariant in [[Knot theory]] | + | * with an R-matrix satisfying the YBE, we obtain a representation of the [[Braid group]], which then gives a link invariant in [[Knot theory]] |
| − | * [[R-matrix]] | + | * [[R-matrix]] |
| 75번째 줄: | 75번째 줄: | ||
==YBE for vertex models== | ==YBE for vertex models== | ||
| − | * Yang-Baxter equation | + | * Yang-Baxter equation |
| − | * conditions satisfied by the Boltzmann weights of vertex models | + | * conditions satisfied by the Boltzmann weights of vertex models |
| − | * has been called the star-triangle relation | + | * has been called the star-triangle relation |
2013년 2월 9일 (토) 01:56 판
introduction
- roles in the following fields
- exact solvability of many models is explained by commuting transfer matrices
- manifestations of Yang-Baxter equation
- \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
- for vertex models, YBE becomes the star-triangle relation
- see [Baxter1995] for a historical account
Yang and Baxter
- [Yang1967] interacting particles with potential
- Bethe ansatz gave rise to an equation
- [Baxter1972] considered the problem of eight-vertex model and quantum XYZ model
- commutation of transfer matrices
Bethe ansatz
- Bethe ansatz amplitude
integrability of a model
- in the space of couplings a submanifold exists, such as that the transfer matrices corresponding to any two points P and P' on it commute
- characterized by a set of equations on the Boltzmann weights
- this set of equations is called the Yang-Baxter equation
- solutions to Yang-Baxter equation can lead to a construction of integrable models
transfer matrix
- borrowed from transfer matrix in statistical mechanics
- transfer matrix is builtup from matrices of Boltzmann weights
- we need the transfer matrices coming from different set of Boltzman weights commute
- partition function = trace of power of transfer matrices
- so the problem of solving the model is reduced to the computation of this trace
R-matrix
- we make a matrix from the Boltzmann weights
- if we can find an R-matrix, then it implies the existence of a set of Boltzmann weights which give exactly solvable models
- that is why we care about the quantum groups
- spectral parameters
- anistropy parameters
- with an R-matrix satisfying the YBE, we obtain a representation of the Braid group, which then gives a link invariant in Knot theory
- R-matrix
YBE for vertex models
- Yang-Baxter equation
- conditions satisfied by the Boltzmann weights of vertex models
- has been called the star-triangle relation
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Yang–Baxter_equation
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(첨부파일로 올릴것)
books
- Knots and physics
- Louis H. Kauffman
- Quantum Groups in Two-Dimensional Physics
- Yang-Baxter Equations, Conformal Invariance And Integrability In Statistical Mechanics And Field Theory
- http://gigapedia.info/1/knots+physics
- http://gigapedia.info/1/two-dimensional+physics
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
articles
- Introduction to the Yang-Baxter equation
- Jimbo
- [Baxter1995]Solvable models in statistical mechanics, from Onsager onward
- Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995
- [Baxter1972]Partition Function of the Eight-Vertex Lattice Model
- Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972
- Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972
- [Yang1967]Some exact results for the many-body problem in one dimension with repulsive delta-function interaction
- C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/10.1007/BF02183337