"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| − | * roles in | + | * most important roles in [[Integrable systems and solvable models]] |
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* at the heart of [[quantum groups]] | * at the heart of [[quantum groups]] | ||
| + | * exact solvability of many models is explained by commuting transfer matrices | ||
* in 1+1D S-matrix theory, the YBE is the condition for consistent factorization of the multiparticle S-matrix into two-particle factors | * in 1+1D S-matrix theory, the YBE is the condition for consistent factorization of the multiparticle S-matrix into two-particle factors | ||
** see [[Exact S-matrices in ATFT]] | ** see [[Exact S-matrices in ATFT]] | ||
| 83번째 줄: | 82번째 줄: | ||
* [[quantum groups]] | * [[quantum groups]] | ||
* [[Yangian]] | * [[Yangian]] | ||
| − | * [[ | + | * [[Proofs and Confirmation]] |
| 95번째 줄: | 94번째 줄: | ||
==books== | ==books== | ||
| − | * [http://gigapedia.com/items:links?id=71502 Knots and physics] | + | * Louis H. Kauffman, [http://gigapedia.com/items:links?id=71502 Knots and physics] |
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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 | ||
2013년 3월 12일 (화) 02:42 판
introduction
- most important roles in Integrable systems and solvable models
- at the heart of quantum groups
- exact solvability of many models is explained by commuting transfer matrices
- in 1+1D S-matrix theory, the YBE is the condition for consistent factorization of the multiparticle S-matrix into two-particle factors
- \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
- 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
- Louis H. Kauffman, Knots and physics
- 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
- Kulish, P. P., N. Yu Reshetikhin, and E. K. Sklyanin. 1981. “Yang-Baxter Equation and Representation Theory: I.” Letters in Mathematical Physics 5 (5) (September 1): 393–403. doi:10.1007/BF02285311. http://dx.doi.org/10.1007/BF02285311
- [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