Extended State Space of the Rational sl(2) Gaudin Model in Terms of Laguerre Polynomials

  • Yu. V. Bezvershenko Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine, Faculty of Physical and Mathematical Sciences, National University of Kyiv-Mohyla Academy
  • P. I. Holod Faculty of Physical and Mathematical Sciences, National University of Kyiv-Mohyla Academy
Keywords: Gaudin model, sl(2) representation theory, Laguerre polynomials

Abstract

We consider the rational Gaudin model with non-zero magnetic field, which physically corresponds to the central spin problem. The space of states is described in terms of separated variables. The states of a spin system are given by rational (up to an exponential factor) functions of these variables on the Lagrangian submanifold. We build a representation of the sl(2) algebra of the model in terms of Laguerre polynomials and formulate the functional Bethe ansatz using it.

References

M. Gaudin, J. Phys. France 37, 1087 (1976).

https://doi.org/10.1051/jphys:0197600370100108700

M. Gaudin, La Fonction d'Onde de Bethe (Masson, Paris, 1983).

E.K. Sklyanin, J. Soviet Math. 47, 2473 (1989).

https://doi.org/10.1007/BF01840429

B.A. Dubrovin, I.M. Krichever, and S.P. Novikov, Encycl. Math. Sci. 4, 173 (1990).

B.A. Dubrovin, Uspekhi Mat. Nauk 36, (1981).

V.P. Kotlyarov, Vopr. Mat. Fiz. Funkts. An. 1, 121 (1976).

A. Varchenko, Comp. Math. 97, 385 (1995).

E. Previato, Duke Math. J. 52, 329 (1985).

https://doi.org/10.1215/S0012-7094-85-05218-4

O. Babelon and D. Talalaev, J. Stat. Mech. 6, 06013 (2007).

https://doi.org/10.1088/1742-5468/2007/06/P06013

B. Jurˇco, J. Math. Phys. 30, 1739 (1989).

https://doi.org/10.1063/1.528262

E.A. Yuzbashyan, B.L. Altshuler, V.B. Kuznetsov, and V.Z. Enolskii, Phys. Rev. B 72, 220503 (2005).

https://doi.org/10.1103/PhysRevB.72.220503

E.A. Yuzbashyan, B.L. Altshuler, V.B. Kuznetsov, and V.Z. Enolskii, J. Phys. A 38, 7831 (2005).

https://doi.org/10.1088/0305-4470/38/36/003

B. Feigin, E. Frenkel, and N. Reshetikhin, Commun. Math. Phys. 166, 27 (1994).

https://doi.org/10.1007/BF02099300

E.D. Belokolos, Ukr. J. Phys. 54, 862 (2009).

A. Faribault, O. El Araby, C. Str¨ater, and V. Gritsev, Phys. Rev. B 83, 235124 (2011).

https://doi.org/10.1103/PhysRevB.83.235124

E.T. Whittaker and G.N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, Cambridge, 1996).

https://doi.org/10.1017/CBO9780511608759

V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989).

https://doi.org/10.1007/978-1-4757-2063-1

G. Szeg¨o, Orthogonal Polynomials (AMS, New York, 1959).

Published
2018-10-11
How to Cite
Bezvershenko, Y., & Holod, P. (2018). Extended State Space of the Rational sl(2) Gaudin Model in Terms of Laguerre Polynomials. Ukrainian Journal of Physics, 58(11), 1084. https://doi.org/10.15407/ujpe58.11.1084
Section
Archive