"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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| − | + | ==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 | ||
| + | ** see [[Exact S-matrices in ATFT]] | ||
| + | * <math>R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)</math> | ||
| + | * for vertex models, YBE becomes the star-triangle relation | ||
| + | * see '''[Baxter1995] '''for a historical account | ||
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| − | + | ==Yang and Baxter== | |
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| − | + | * '''[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 | |
| − | + | ||
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| − | + | ||
| − | + | ==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 | ||
| − | + | ||
| − | + | ||
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| − | + | ==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 | ||
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| − | + | ||
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| − | + | ==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 | ||
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| − | + | ==classical YBE== | |
| − | + | :<math> | |
| + | [X_{12}(u_1-u_2),X_{13}(u_1-u_3)]+[X_{13}(u_1-u_3),X_{23}(u_2-u_3)]+[X_{12}(u_1-u_2),X_{23}(u_2-u_3)]=0 | ||
| + | </math> | ||
| + | |||
| − | + | ==related items== | |
| + | * [[Belavin-Drinfeld theory]] | ||
| + | * [[Quantum groups]] | ||
| + | * [[Yangian]] | ||
| + | * [[Sklyanin algebra]] | ||
| + | * [[Proofs and Confirmation]] | ||
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| − | + | ==computational resource== | |
| − | + | * [[R-matrix]] | |
| − | + | * https://docs.google.com/file/d/0B8XXo8Tve1cxdDk5Rm9DQy1nelk/edit | |
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| + | ==encyclopedia== | ||
* 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] |
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| − | + | ==books== | |
| − | * | + | * 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 | ||
| − | * | + | * knots+physics |
| − | * | + | * two-dimensional+physics |
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| − | + | ==expositions== | |
| − | + | * Nichita, Florin F. “Yang-Baxter Equations, Computational Methods and Applications.” arXiv:1506.03610 [math-Ph], June 11, 2015. http://arxiv.org/abs/1506.03610. | |
| − | + | * [http://mathlab.snu.ac.kr/~top/quantum/seminar/0521.pdf Yang-Baxter equation in Physics], 안창림 | |
| − | + | * http://math.ucr.edu/home/baez/braids/node4.html | |
| − | + | * Perk, Jacques H. H., and Helen Au-Yang. 2006. “Yang-Baxter Equations.” arXiv:math-ph/0606053 (June 20). http://arxiv.org/abs/math-ph/0606053. | |
| + | * '''[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 | ||
| + | * Hietarinta, J. “The Complete Solution to the Constant Quantum Yang-Baxter Equation in Two Dimensions.” In Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, edited by Peter A. Clarkson, 149–54. NATO ASI Series 413. Springer Netherlands, 1993. http://link.springer.com/chapter/10.1007/978-94-011-2082-1_15. | ||
| + | * Jimbo, Michio. 1989. “Introduction to the Yang-Baxter Equation.” International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics 4 (15): 3759–3777. doi:10.1142/S0217751X89001503. http://www.worldscientific.com/doi/abs/10.1142/S0217751X89001503 | ||
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| − | * [http:// | + | ==articles== |
| − | * | + | * Yamazaki, Masahito, and Wenbin Yan. ‘Integrability from 2d N=(2,2) Dualities’. arXiv:1504.05540 [hep-Th, Physics:math-Ph], 21 April 2015. http://arxiv.org/abs/1504.05540. |
| − | * [http:// | + | * Chicherin, D., and S. Derkachov. ‘Matrix Factorization for Solutions of the Yang-Baxter Equation’. arXiv:1502.07923 [hep-Th, Physics:math-Ph], 27 February 2015. http://arxiv.org/abs/1502.07923. |
| − | + | * Chicherin, D., S. E. Derkachov, and V. P. Spiridonov. “New Elliptic Solutions of the Yang-Baxter Equation.” arXiv:1412.3383 [hep-Th, Physics:math-Ph], December 10, 2014. http://arxiv.org/abs/1412.3383. | |
| − | * [http://dx.doi.org/10. | + | * Chicherin, D., S. E. Derkachov, and V. P. Spiridonov. “From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation.” arXiv:1411.7595 [hep-Th, Physics:math-Ph], November 27, 2014. http://arxiv.org/abs/1411.7595. |
| − | * | + | * Hietarinta, Jarmo. “Solving the Two‐dimensional Constant Quantum Yang–Baxter Equation.” Journal of Mathematical Physics 34, no. 5 (May 1, 1993): 1725–56. doi:10.1063/1.530185. |
| − | * [http://dx.doi.org/10.1006/aphy.2000.6010 Partition Function of the Eight-Vertex | + | * Hietarinta, Jarmo. “All Solutions to the Constant Quantum Yang-Baxter Equation in Two Dimensions.” Physics Letters A 165, no. 3 (May 18, 1992): 245–51. doi:10.1016/0375-9601(92)90044-M. |
| − | ** Baxter, Rodney , | + | * Belavin, A. A., and V. G. Drinfel’d. 1982. “Solutions of the Classical Yang - Baxter Equation for Simple Lie Algebras.” Functional Analysis and Its Applications 16 (3) (July 1): 159–180. doi:[http://dx.doi.org/10.1007/BF01081585 10.1007/BF01081585]. |
| + | * 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]'''[http://dx.doi.org/10.1006/aphy.2000.6010 Partition Function of the Eight-Vertex Lattice Model] | ||
| + | ** Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972 | ||
| + | * '''[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] | ||
| + | ** C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315 | ||
| − | * | + | ==questions== |
| − | + | * http://mathoverflow.net/questions/5103/solutions-of-the-quantum-yang-baxter-equation | |
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| − | + | [[분류:개인노트]] | |
| − | + | [[분류:integrable systems]] | |
| − | + | [[분류:math and physics]] | |
| − | + | [[분류:quantum groups]] | |
| − | + | [[분류:migrate]] | |
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| − | + | ==메타데이터== | |
| − | + | ===위키데이터=== | |
| − | + | * ID : [https://www.wikidata.org/wiki/Q4476530 Q4476530] | |
| − | + | ===Spacy 패턴 목록=== | |
| − | + | * [{'LOWER': 'yang'}, {'OP': '*'}, {'LOWER': 'baxter'}, {'LEMMA': 'equation'}] | |
| − | + | * [{'LOWER': 'r'}, {'OP': '*'}, {'LEMMA': 'matrix'}] | |
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2021년 2월 17일 (수) 01:28 기준 최신판
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
- \(R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)\)
- 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
classical YBE
\[ [X_{12}(u_1-u_2),X_{13}(u_1-u_3)]+[X_{13}(u_1-u_3),X_{23}(u_2-u_3)]+[X_{12}(u_1-u_2),X_{23}(u_2-u_3)]=0 \]
computational resource
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
- Nichita, Florin F. “Yang-Baxter Equations, Computational Methods and Applications.” arXiv:1506.03610 [math-Ph], June 11, 2015. http://arxiv.org/abs/1506.03610.
- Yang-Baxter equation in Physics, 안창림
- http://math.ucr.edu/home/baez/braids/node4.html
- Perk, Jacques H. H., and Helen Au-Yang. 2006. “Yang-Baxter Equations.” arXiv:math-ph/0606053 (June 20). http://arxiv.org/abs/math-ph/0606053.
- [Baxter1995] BaxterSolvable models in statistical mechanics, from Onsager onward, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995
- Hietarinta, J. “The Complete Solution to the Constant Quantum Yang-Baxter Equation in Two Dimensions.” In Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, edited by Peter A. Clarkson, 149–54. NATO ASI Series 413. Springer Netherlands, 1993. http://link.springer.com/chapter/10.1007/978-94-011-2082-1_15.
- Jimbo, Michio. 1989. “Introduction to the Yang-Baxter Equation.” International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics 4 (15): 3759–3777. doi:10.1142/S0217751X89001503. http://www.worldscientific.com/doi/abs/10.1142/S0217751X89001503
articles
- Yamazaki, Masahito, and Wenbin Yan. ‘Integrability from 2d N=(2,2) Dualities’. arXiv:1504.05540 [hep-Th, Physics:math-Ph], 21 April 2015. http://arxiv.org/abs/1504.05540.
- Chicherin, D., and S. Derkachov. ‘Matrix Factorization for Solutions of the Yang-Baxter Equation’. arXiv:1502.07923 [hep-Th, Physics:math-Ph], 27 February 2015. http://arxiv.org/abs/1502.07923.
- Chicherin, D., S. E. Derkachov, and V. P. Spiridonov. “New Elliptic Solutions of the Yang-Baxter Equation.” arXiv:1412.3383 [hep-Th, Physics:math-Ph], December 10, 2014. http://arxiv.org/abs/1412.3383.
- Chicherin, D., S. E. Derkachov, and V. P. Spiridonov. “From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation.” arXiv:1411.7595 [hep-Th, Physics:math-Ph], November 27, 2014. http://arxiv.org/abs/1411.7595.
- Hietarinta, Jarmo. “Solving the Two‐dimensional Constant Quantum Yang–Baxter Equation.” Journal of Mathematical Physics 34, no. 5 (May 1, 1993): 1725–56. doi:10.1063/1.530185.
- Hietarinta, Jarmo. “All Solutions to the Constant Quantum Yang-Baxter Equation in Two Dimensions.” Physics Letters A 165, no. 3 (May 18, 1992): 245–51. doi:10.1016/0375-9601(92)90044-M.
- Belavin, A. A., and V. G. Drinfel’d. 1982. “Solutions of the Classical Yang - Baxter Equation for Simple Lie Algebras.” Functional Analysis and Its Applications 16 (3) (July 1): 159–180. doi:10.1007/BF01081585.
- 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
- [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
questions
메타데이터
위키데이터
- ID : Q4476530
Spacy 패턴 목록
- [{'LOWER': 'yang'}, {'OP': '*'}, {'LOWER': 'baxter'}, {'LEMMA': 'equation'}]
- [{'LOWER': 'r'}, {'OP': '*'}, {'LEMMA': 'matrix'}]