"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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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> | ||
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* Introduction to the Yang-Baxter equation<br> | * Introduction to the Yang-Baxter equation<br> | ||
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2012년 10월 28일 (일) 16:11 판
introduction
- roles in the following fields
- integrable systems
- exactly solvable statistical models
- quantum groups
- conformal field theory
- topological quantum field theory
- braid groups
- 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=
- Louis H. Kauffman
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
- [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
- Jimbo
- Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995
- Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972
- C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315