"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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==related items== | ==related items== | ||
| − | + | * [[quantum groups]] | |
| − | + | * [[Yangian]] | |
==encyclopedia== | ==encyclopedia== | ||
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
* [http://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation http://en.wikipedia.org/wiki/Yang–Baxter_equation] | * [http://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation http://en.wikipedia.org/wiki/Yang–Baxter_equation] | ||
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==books== | ==books== | ||
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* [http://www.amazon.com/Quantum-Two-Dimensional-Cambridge-Monographs-Mathematical/dp/0521460654 Quantum Groups in Two-Dimensional Physics] | * [http://www.amazon.com/Quantum-Two-Dimensional-Cambridge-Monographs-Mathematical/dp/0521460654 Quantum Groups in Two-Dimensional Physics] | ||
* Yang-Baxter Equations, Conformal Invariance And Integrability In Statistical Mechanics And Field Theory | * Yang-Baxter Equations, Conformal Invariance And Integrability In Statistical Mechanics And Field Theory | ||
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| − | * | + | * two-dimensional+physics |
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| − | + | ==expositions== | |
| + | * http://math.ucr.edu/home/baez/braids/node4.html | ||
| + | * Jimbo, Introduction to the Yang-Baxter equation | ||
| + | * '''[Baxter1995]''' Baxter[http://dx.doi.org/10.1007/BF02183337 Solvable models in statistical mechanics, from Onsager onward], Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995 | ||
==articles== | ==articles== | ||
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* '''[Baxter1972]'''[http://dx.doi.org/10.1006/aphy.2000.6010 Partition Function of the Eight-Vertex Lattice Model]<br> | * '''[Baxter1972]'''[http://dx.doi.org/10.1006/aphy.2000.6010 Partition Function of the Eight-Vertex Lattice Model]<br> | ||
** 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> | ||
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** C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315 | ** C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315 | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:integrable systems]] | [[분류:integrable systems]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
2013년 3월 8일 (금) 12:53 판
introduction
- roles in the following fields
- exact solvability of many models is explained by commuting transfer matrices
- at the heart of quantum groups
- 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
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
- knots+physics
- two-dimensional+physics
expositions
- http://math.ucr.edu/home/baez/braids/node4.html
- Jimbo, Introduction to the Yang-Baxter equation
- [Baxter1995] BaxterSolvable models in statistical mechanics, from Onsager onward, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995
articles
- [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