"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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| − | + | ==introduction== | |
* exact solvability of many models is explained by commuting transfer matrices<br> | * exact solvability of many models is explained by commuting transfer matrices<br> | ||
| 6번째 줄: | 6번째 줄: | ||
* <math>R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}</math><br> | * <math>R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}</math><br> | ||
* for vertex models, YBE becomes the star-triangle relation<br> | * for vertex models, YBE becomes the star-triangle relation<br> | ||
| − | * | + | * see '''[Baxter1995] '''for a historical account<br> |
| − | + | ||
| − | + | ||
| − | + | ==Yang and Baxter== | |
| − | * '''[Yang1967]''' | + | * '''[Yang1967]''' [[interacting particles with potential]]<br> |
| − | ** Bethe ansatz | + | ** Bethe ansatz gave rise to an equation <br> |
| − | * '''[Baxter1972] | + | * '''[Baxter1972] '''considered the problem of [[eight-vertex model and quantum XYZ model]]<br> |
** commutation of transfer matrices<br> | ** commutation of transfer matrices<br> | ||
| − | + | ||
| − | + | ||
| − | + | ==Bethe ansatz== | |
| − | * [[Bethe ansatz]] | + | * [[Bethe ansatz]] amplitude<br> |
| − | + | ||
| − | + | ||
| − | + | ==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<br> | * 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<br> | ||
| 38번째 줄: | 38번째 줄: | ||
* solutions to Yang-Baxter equation can lead to a construction of integrable models<br> | * solutions to Yang-Baxter equation can lead to a construction of integrable models<br> | ||
| − | + | ||
| − | + | ||
| − | + | ==transfer matrix== | |
| − | * borrowed | + | * borrowed from [[transfer matrix in statistical mechanics]]<br> |
| − | * transfer matrix is builtup from matrices of | + | * transfer matrix is builtup from matrices of Boltzmann weights<br> |
| − | * we need the transfer matrices coming from different set of Boltzman weights | + | * we need the transfer matrices coming from different set of Boltzman weights commute <br> |
| − | * partition function = | + | * partition function = trace of power of transfer matrices<br> |
* so the problem of solving the model is reduced to the computation of this trace<br> | * so the problem of solving the model is reduced to the computation of this trace<br> | ||
| − | + | ||
| − | + | ||
| − | + | ==R-matrix== | |
* we make a matrix from the Boltzmann weights<br> | * we make a matrix from the Boltzmann weights<br> | ||
| − | * if we can find an R-matrix, then it implies the existence of | + | * if we can find an R-matrix, then it implies the existence of a set of Boltzmann weights which give exactly solvable models<br> |
* that is why we care about the quantum groups<br> | * that is why we care about the quantum groups<br> | ||
* spectral parameters<br> | * spectral parameters<br> | ||
| 64번째 줄: | 64번째 줄: | ||
* [[R-matrix]]<br> | * [[R-matrix]]<br> | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ==YBE for vertex models== | |
* Yang-Baxter equation<br> | * Yang-Baxter equation<br> | ||
* conditions satisfied by the Boltzmann weights of vertex models<br> | * conditions satisfied by the Boltzmann weights of vertex models<br> | ||
| − | * has been called | + | * has been called the star-triangle relation<br> |
| − | + | ||
| − | + | ||
| − | + | ==related items== | |
2012년 10월 14일 (일) 08:21 판
introduction
- exact solvability of many models is explained by commuting transfer matrices
- manifestations of Yang-Baxter equation
- factorizable S-matrix
- factorizable S-matrix
- \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
- 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
- Bethe ansatz gave rise to an equation
- [Baxter1972] considered the problem of eight-vertex model and quantum XYZ model
- commutation of transfer matrices
- 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
- 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