"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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(피타고라스님이 이 페이지의 이름을 Yang-Baxter equation (YBE)로 바꾸었습니다.) |
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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 | + | * see '''[Baxter1995] '''for a historical account<br> |
14번째 줄: | 14번째 줄: | ||
<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">Yang and Baxter</h5> | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">Yang and Baxter</h5> | ||
− | * | + | * '''[Yang1967]''' [[interacting particles with potential]]<br> |
− | * | + | ** Bethe ansatz gave rise to an equation <br> |
+ | * '''[Baxter1972] '''considered the problem of [[eight-vertex model and quantum XYZ model]]<br> | ||
+ | ** commutation of transfer matrices<br> | ||
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+ | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">Bethe ansatz</h5> | ||
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+ | * [[Bethe ansatz]] amplitude<br> | ||
+ | * <br> | ||
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80번째 줄: | 93번째 줄: | ||
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
− | * http://en.wikipedia.org/wiki/Yang | + | * [http://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation http://en.wikipedia.org/wiki/Yang–Baxter_equation] |
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
93번째 줄: | 106번째 줄: | ||
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">books</h5> | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">books</h5> | ||
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* [http://gigapedia.com/items:links?id=71502 Knots and physics]<br> | * [http://gigapedia.com/items:links?id=71502 Knots and physics]<br> | ||
** Louis H. Kauffman | ** Louis H. Kauffman | ||
110번째 줄: | 122번째 줄: | ||
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">articles</h5> | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">articles</h5> | ||
+ | * <br> | ||
* '''[Baxter1995]'''[http://dx.doi.org/10.1007/BF02183337 Solvable models in statistical mechanics, from Onsager onward]<br> | * '''[Baxter1995]'''[http://dx.doi.org/10.1007/BF02183337 Solvable models in statistical mechanics, from Onsager onward]<br> | ||
** Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995 | ** Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995 | ||
− | + | * '''[Baxter1972]'''[http://dx.doi.org/10.1006/aphy.2000.6010 Partition Function of the Eight-Vertex Lattice Model]<br> | |
− | * [ | ||
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** Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972<br> | ** Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972<br> | ||
− | * <br>[http://dx.doi.org/10.1103/PhysRevLett.19.1312 Some exact results for the many-body problem in one dimension with repulsive delta-function interaction]<br> | + | * <br> |
+ | * '''[Yang1967]'''[http://dx.doi.org/10.1103/PhysRevLett.19.1312 Some exact results for the many-body problem in one dimension with repulsive delta-function interaction]<br> | ||
** C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315 | ** C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315 | ||
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* http://www.ams.org/mathscinet | * http://www.ams.org/mathscinet |
2010년 8월 3일 (화) 16:00 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
Bethe ansatz
YBE for vertex models
- Yang-Baxter equation
- conditions satisfied by the Boltzmann weights of vertex models
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
-
- [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