"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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2010년 8월 3일 (화) 15:44 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 history
Yang and Baxter
- Yang199
- Baxter considered the problem of eight-vertex model and quantum XYZ 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
- 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
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(첨부파일로 올릴것)
books
- 찾아볼 수학책
- 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
- Integrable theories, Yang-Baxter algebras and quantum groups: An overview
- H. J. De Vega, Lecture Notes in Physics, Volume 382, 1991
- Yang-Baxter algebras, integrable theories and Betre Ansatz
- H. J. De Vega, Volume: 4, Issue: 5(1990) pp. 735-801
- Yang-Baxter algebras, integrable theories and quantum groups
- H. J. De Vega, Volume: 4, Issue: 10(1989) pp. 2371-2463
- 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
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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
- One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System
- C. N. Yang, C. P. Yang, 1966
- One-dimensional chain of anisotropic spin-spin interactions
- C. N. Yang, C. P. Yang, 1966
- 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