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Combinatorics of Generalized Bethe Equations

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Abstract

A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over \({\mathbb{Z}^M}\), and on the other hand, they count integer points in certain M-dimensional polytopes.

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References

  1. Bethe H.: On the theory of metals: eigenvalues and eigenfunctions of a linear chain of atoms. Z. Phys. 71, 205–226 (1931)

    Article  ADS  Google Scholar 

  2. Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  3. Bogoliubov, N.M., Izergin, A.G., Korepin, V.E.: Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1993). ISBN: 9780521586467

  4. Faddeev, L.D.: How Algebraic Bethe Ansatz Works for Integrable Model. In: Quantum Symmetries/Symetries Quantiques: Proceedings of the Les Houches Summer School, Session Lxiv, Les Houches, France (1995)

  5. Reshetikhin N.: A method of functional equations in the theory of exactly solvable quantum systems. Lett. Math. Phys. 7, 205–213 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  6. Tarasov V.O.: Algebraic Bethe ansatz for the Izergin–Korepin R-matrix. Theor. Math. Phys. 76, 793–803 (1988)

    Article  MathSciNet  Google Scholar 

  7. Kazakov V.A., Marshakov A., Minahan J.A., Zarembo K.: Classical/quantum integrability in AdS/CFT. J. High Energy Phys. 05, 024 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  8. Bargheer T., Beisert N., Loebbert F.: Long-range deformations for integrable spin chains. J. Phys. A Math. Gen. 42, 285205 (2009)

    Article  MathSciNet  Google Scholar 

  9. Faddeev L.D., Takhtadzhan L.A.: Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model. J. Math. Sci. 24, 241–267 (1984)

    Article  MATH  Google Scholar 

  10. Kirillov A.N., Reshetikhin N.Yu.: The Yangians, Bethe ansatz and combinatorics. Lett. Math. Phys. 12, 199–208 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison-Wesley, Reading (1994)

  12. Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997)

  13. Hilton P., Pedersen J.: Catalan numbers, their generalizations and their uses. Math. Intell. 13, 64–75 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Stanley, R.P.: Catalan addendum. http://www-math.mit.edu/~rstan/ec/catadd.pdf

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Correspondence to Evgeny K. Sklyanin.

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E. K. Sklyanin work supported by EPSRC Grant EP/H000054/1.

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Kozlowski, K.K., Sklyanin, E.K. Combinatorics of Generalized Bethe Equations. Lett Math Phys 103, 1047–1077 (2013). https://doi.org/10.1007/s11005-013-0630-9

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  • DOI: https://doi.org/10.1007/s11005-013-0630-9

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