Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 047, 27 pages      arXiv:2006.01417      https://doi.org/10.3842/SIGMA.2020.047

New Separation of Variables for the Classical XXX and XXZ Heisenberg Spin Chains

Guido Magnano a and Taras Skrypnyk ab
a) Università degli Studi di Torino, via Carlo Alberto 10, 10123, Torino, Italia
b) Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine

Received December 20, 2019, in final form May 16, 2020; Published online June 02, 2020

Abstract
We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix. We show that for the case of such twist matrices one can interchange the role of classical separating functions $A(u)$ and $B(u)$ and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin. We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.

Key words: integrable spin chains; quadratic Sklyanin brackets; separation of variables.

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References

  1. Adams M.R., Harnad J., Hurtubise J., Darboux coordinates and Liouville-Arnol'd integration in loop algebras, Comm. Math. Phys. 155 (1993), 385-413, arXiv:hep-th/9210089.
  2. Al'ber S.I., Investigation of equations of Korteweg-de Vries type by the method of recurrence relations, J. London Math. Soc. 19 (1979), 467-480.
  3. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
  4. Diener P., Dubrovin B., Algebraic-geometrical Darboux coordinates in $R$-matrix formalism, Preprint SISSA-88-94-FM, 1994.
  5. Dubrovin B., Skrypnyk T., Separation of variables for linear Lax algebras and classical $r$-matrices, J. Math. Phys. 59 (2018), 091405, 39 pages.
  6. Dubrovin B., Skrypnyk T., Separation of variables for quadratic algebras and skew-symmetric classical $r$-matrices, J. Math. Phys. 60 (2019), 093506, 30 pages.
  7. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987.
  8. Gekhtman M.I., Separation of variables in the classical ${\rm SL}(N)$ magnetic chain, Comm. Math. Phys. 167 (1995), 593-605.
  9. Jurčo B., Classical Yang-Baxter equations and quantum integrable systems, J. Math. Phys. 30 (1989), 1289-1293.
  10. Kitanine N., Maillet J.M., Niccoli G., Terras V., The open XXX spin chain in the SoV framework: scalar product of separate states, J. Phys. A: Math. Theor. 50 (2017), 224001, 35 pages, arXiv:1606.06917.
  11. Kitanine N., Maillet J.M., Niccoli G., Terras V., The open XXZ spin chain in the SoV framework: scalar product of separate states, J. Phys. A: Math. Theor. 51 (2018), 485201, 46 pages, arXiv:1807.05197.
  12. Lax P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490.
  13. Magri F., The Kowalewski top revisited, in Integrable systems and Algebraic Geometry, Vol. 1, London Mathematical Society Lecture Note Series, Vol. 458, Editors R. Donagi, T. Shaska, Cambridge University Press, Cambridge, 2020, 329-355, arXiv:1809.01879.
  14. Maillet J.M., Niccoli G., On quantum separation of variables, J. Math. Phys. 59 (2018), 091417, 47 pages, arXiv:1807.11572.
  15. Reshetikhin N.Yu., Faddeev L.D., Hamiltonian structures for integrable models of field theory, Theoret. and Math. Phys. 56 (1983), 847-862.
  16. Roubtsov V., Skrypnyk T., Compatible Poisson brackets, quadratic Poisson algebras and classical $r$-matrices, in Differential Equations: Geometry, Symmetries and Integrability, Abel Symp., Vol. 5, Springer, Berlin, 2009, 311-333.
  17. Ryan P., Volin D., Separated variables and wave functions for rational $\mathfrak{gl}(N)$ spin chains in the companion twist frame, J. Math. Phys. 60 (2019), 032701, 23 pages, arXiv:1810.10996.
  18. Scott D.R.D., Classical functional Bethe ansatz for ${\rm SL}(N)$: separation of variables for the magnetic chain, J. Math. Phys. 35 (1994), 5831-5843, arXiv:hep-th/9403030.
  19. Sklyanin E.K., On complete integrability of the Landau-Lifshitz equation, Preprint LOMI E-3, 1979.
  20. Sklyanin E.K., The quantum Toda chain, in Nonlinear Equations in Classical and Quantum Field Theory (Meudon/Paris, 1983/1984), Lecture Notes in Phys., Vol. 226, Springer, Berlin, 1985, 196-233.
  21. Sklyanin E.K., Poisson structure of classical $XXZ$-chain, J. Soviet Math. 46 (1989), 2104-2111.
  22. Sklyanin E.K., Separation of variables in the Gaudin model, J. Soviet Math. 47 (1989), 2473-2488.
  23. Sklyanin E.K., Separation of variables in the classical integrable ${\rm SL}(3)$ magnetic chain, Comm. Math. Phys. 150 (1992), 181-191, arXiv:hep-th/9211126.
  24. Sklyanin E.K., Separation of variables - new trends, Progr. Theoret. Phys. Suppl. 118 (1995), 35-60, arXiv:solv-int/9504001.
  25. Veselov A.P., Novikov S.P., On Poisson brackets compatible with algebraic geometry and Korteweg-de Vries dynamics on the space of finite-zone potentials, Soviet Math. Dokl. 26 (1982), 533-537.

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