Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles
Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 2, pp. 235-249 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric (hyperbolic) functions.
Keywords: six-vertex model, Ruijsenaars–Schneider system of particles, quantum–classical correspondence.
@article{TMF_2017_192_2_a4,
     author = {A. V. Zabrodin and A. V. Zotov and A. N. Liashyk and D. S. Rudneva},
     title = {Asymmetric six-vertex model and the~classical {Ruijsenaars{\textendash}Schneider} system of particles},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {235--249},
     year = {2017},
     volume = {192},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a4/}
}
TY  - JOUR
AU  - A. V. Zabrodin
AU  - A. V. Zotov
AU  - A. N. Liashyk
AU  - D. S. Rudneva
TI  - Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2017
SP  - 235
EP  - 249
VL  - 192
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a4/
LA  - ru
ID  - TMF_2017_192_2_a4
ER  - 
%0 Journal Article
%A A. V. Zabrodin
%A A. V. Zotov
%A A. N. Liashyk
%A D. S. Rudneva
%T Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2017
%P 235-249
%V 192
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a4/
%G ru
%F TMF_2017_192_2_a4
A. V. Zabrodin; A. V. Zotov; A. N. Liashyk; D. S. Rudneva. Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles. Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 2, pp. 235-249. http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a4/

[1] A. Alexandrov, V. Kazakov, S. Leurent, Z. Tsuboi, A. Zabrodin, “Classical tau-function for quantum spin chains”, JHEP, 09 (2013), 064, 64 pp. | DOI | MR

[2] A. Alexandrov, S. Leurent, Z. Tsuboi, A. Zabrodin, “The master $T$-operator for the Gaudin model and the KP hierarchy”, Nucl. Phys. B, 883 (2014), 173–223 | DOI | MR

[3] A. Gorsky, A. Zabrodin, A. Zotov, “Spectrum of quantum transfer matrices via classical many-body systems”, JHEP, 01 (2014), 070, 27 pp. | DOI | MR

[4] A. Zabrodin, “Quantum spin chains and integrable many-body systems of classical mechanics”, Nonlinear Mathematical Physics and Natural Hazards (Sofia, Bulgaria, 28 November – 2 December, 2013), Springer Proceedings in Physics, 163, eds. B. Aneva, M. Kouteva-Guentcheva, Springer, Cham, 2015, 29–48 | DOI | Zbl

[5] Z. Tsuboi, A. Zabrodin, A. Zotov, “Supersymmetric quantum spin chains and classical integrable systems”, JHEP, 05 (2015), 086, 42 pp. | DOI | MR

[6] E. Mukhin, V. Tarasov, A. Varchenko, “KZ characteristic variety as the zero set of classical Calogero–Moser Hamiltonians”, SIGMA, 8 (2012), 072, 11 pp. | DOI | MR | Zbl

[7] N. Nekrasov, A. Rosly, S. Shatashvili, “Darboux coordinates, Yang–Yang functional, and gauge theory”, Nucl. Phys. B Proc. Suppl., 216:1 (2011), 69–93 | DOI | MR

[8] A. V. Zabrodin, “Upravlyayuschii $T$-operator dlya vershinnykh modelei s trigonometricheskimi $R$-matritsami kak klassicheskaya $\tau$-funktsiya”, TMF, 174:1 (2013), 59–76 | DOI | DOI | MR | Zbl

[9] M. Beketov, A. Liashyk, A. Zabrodin, A. Zotov, “Trigonometric version of quantum-classical duality in integrable systems”, Nucl. Phys. B, 903 (2016), 150–163 | DOI | MR | Zbl

[10] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982 | MR | Zbl

[11] N. Reshetikhin, “Lectures on integrability of the six-vertex model”, Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing (École d'été de Physique des Houches, session LXXXIX, 30 June – 1 August, 2008), eds. J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban, L. F. Cugliandolo, Oxford Univ. Press, Oxford, 2010, 197–266 | MR | Zbl

[12] B. Brubaker, D. Bump, S. Friedberg, “Schur Polynomials and the Yang–Baxter equation”, Commun. Math. Phys., 308:2 (2011), 281–301 | DOI | MR | Zbl

[13] V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge, 1993 | DOI | MR

[14] F. H. L. Essler, V. Rittenberg, “Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries”, J. Phys. A: Math. Gen., 29:13 (1996), 3375–3407 | DOI | MR | Zbl

[15] L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi i $XYZ$ model Geizenberga”, UMN, 34:5(209) (1979), 13–63 | DOI | MR

[16] S. N. M. Ruijsenaars, H. Schneider, “A new class of integrable systems and its relation to solitons”, Ann. Phys., 170:2 (1986), 370–405 | DOI | MR

[17] G. Aminov, S. Arthamonov, A. Smirnov, A. Zotov, “Rational top and its classical $R$-matrix”, J. Phys. A: Math. Theor., 47:30 (2014), 305207, 19 pp. | DOI | MR | Zbl

[18] A. Levin, M. Olshanetsky, A. Zotov, “Relativistic classical integrable tops and quantum $R$-matrices”, JHEP, 07 (2014), 012, 38 pp. | DOI

[19] A. Antonov, K. Hasegawa, A. Zabrodin, “On trigonometric intertwining vectors and non-dynamical $R$-matrix for the Ruijsenaars model”, Nucl. Phys. B, 503:3 (1997), 747–770 | DOI | MR

[20] A. Matsuo, “Integrable connections related to zonal spherical function”, Invent. Math., 110:1 (1992), 95–121 | DOI | MR | Zbl

[21] I. Cherednik, “Integration of quantum many-body problems by affine Knizhnik–Zamolodchikov equations”, Adv. Math., 106:1 (1994), 65–95 | DOI | MR | Zbl