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Semi-classical Scalar Products in the Generalised SU(2) Model

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Book cover Lie Theory and Its Applications in Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 111))

Abstract

In these notes we review the field-theoretical approach to the computation of the scalar product of multi-magnon states in the Sutherland limit where the magnon rapidities condense into one or several macroscopic arrays. We formulate a systematic procedure for computing the 1∕M expansion of the on-shell/off-shell scalar product of M-magnon states in the generalised integrable model with SU(2)-invariant rational R-matrix. The coefficients of the expansion are obtained as multiple contour integrals in the rapidity plane.

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Notes

  1. 1.

    This is a particular case of the Drinfeld polynomial P 1(u) [27] when all spins along the chain are equal to 1∕2.

  2. 2.

    This property is particular for the SU(2) model. The the inner product in the SU(n) model is a determinant only for a restricted class of states [29].

  3. 3.

    The case considered in [21] was that of the periodic inhomogeneous XXX1∕2 spin chain of length L, but the proof given there is trivially extended to the generalised SU(2) model.

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Acknowledgements

The author thanks E. Bettelheim, N. Gromov, T. McLoughlin and S. Shatashvili and for valuable discussions. This work has been supported by European Programme IRSES UNIFY (Grant No. 269217).

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Authors and Affiliations

  1. Institut de Physique Théorique, CNRS-URA 2306, C.E.A.-Saclay, 91191, Gif-sur-Yvette, France

    Ivan Kostov

  2. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussée, 1784, Sofia, Bulgaria

    Ivan Kostov

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Correspondence to Ivan Kostov .

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  1. Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria

    Vladimir Dobrev

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Kostov, I. (2014). Semi-classical Scalar Products in the Generalised SU(2) Model. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 111. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55285-7_7

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