Skip to main content
SpringerLink
Log in

Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Alexandrov, V. Kazakov, S. Leurent, Z. Tsuboi, and A. Zabrodin, “Classical tau-function for quantum spin chains,” JHEP, 09, 064 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. A. Gorsky, A. Zabrodin, and A. Zotov, “Spectrum of quantum transfer matrices via classical many-body systems,” JHEP, 01, 070 (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. A. Zabrodin, “Quantum spin chains and integrable many-body systems of classical mechanics,” in: Nonlinear Mathematical Physics and Natural Hazards (Springer Proc. Phys., Vol. 163, B. Aneva and M. Kouteva-Guentcheva, eds.), Springer, Cham (2015), pp. 29–48.

    Google Scholar 

  5. Z. Tsuboi, A. Zabrodin, and A. Zotov, “Supersymmetric quantum spin chains and classical integrable systems,” JHEP, 05, 086 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  6. E. Mukhin, V. Tarasov, and A. Varchenko, “KZ characteristic variety as the zero set of classical Calogero–Moser Hamiltonians,” SIGMA, 8, 072 (2012).

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. A. V. Zabrodin, “The master T-operator for vertex models with trigonometric R-matrices as a classical τ- function,” Theor. Math. Phys., 174, 52–67 (2013).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982).

    MATH  Google Scholar 

  11. N. Reshetikhin, “Lectures on integrability of the six-vertex model,” in: Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing (École d’été de Physique des Houches, Session 89, 30 June–1 August 2008, J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban, and L. F. Cugliandolo, eds.), Oxford Univ. Press, Oxford (2010), pp. 197–266.

    Google Scholar 

  12. B. Brubaker, D. Bump, and S. Friedberg, “Schur Polynomials and the Yang–Baxter equation,” Commun. Math. Phys., 308, 281–301 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge (1993).

    Book  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. L. A. Takhtadzhyan and L. D. Faddeev, “The quantum method of the inverse problem and the Heisenberg XYZ model,” Russ. Math. Surveys, 34, 11–68 (1979).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. G. Aminov, S. Arthamonov, A. Smirnov, and A. Zotov, “Rational top and its classical R-matrix,” J. Phys. A: Math. Theor., 47, 305207 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Levin, M. Olshanetsky, and A. Zotov, “Relativistic classical integrable tops and quantum R-matrices,” JHEP, 07, 012 (2014).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. A. Matsuo, “Integrable connections related to zonal spherical function,” Invent. Math., 110, 95–121 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. I. Cherednik, “Integration of quantum many-body problems by affine Knizhnik–Zamolodchikov equations,” Adv. Math., 106, 65–95 (1994).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. V. Zabrodin & A. V. Zotov

  2. National Research University Higher School of Economics, Moscow, Russia

    A. V. Zabrodin, A. N. Liashyk & D. S. Rudneva

  3. Skolkovo Institute of Science and Technology, Moscow, Russia

    A. V. Zabrodin & A. N. Liashyk

  4. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia

    A. V. Zotov

  5. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    A. V. Zotov

  6. Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine

    A. N. Liashyk

  7. University of California, Santa Barbara, California, USA

    D. S. Rudneva

Authors
  1. A. V. Zabrodin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Zotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. N. Liashyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. S. Rudneva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Zabrodin.

Additional information

This research was supported by the Russian Federation program for supporting leading universities (Project 5-100).

The research of A. N. Liashyk was supported by the joint NASU-CNRS Project F14-2016.

The research of A. V. Zabrodin was supported in part by the Russian Foundation for Basic Research (Grant No. 14-02-00627).

The research of A. V. Zotov was supported in part by the Russian Foundation for Basic Research (Grant Nos. 14-01-00860 and 15-51-52031 HHC a ).

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 192, No. 2, pp. 235–249, August, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zabrodin, A.V., Zotov, A.V., Liashyk, A.N. et al. Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles. Theor Math Phys 192, 1141–1153 (2017). https://doi.org/10.1134/S0040577917080050

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577917080050

Keywords

Search

Navigation