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Nested Algebraic Bethe Ansatz for Open Spin Chains with Even Twisted Yangian Symmetry

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Abstract

We present a nested algebraic Bethe ansatz for a one-dimensional open spin chain whose boundary quantum spaces are irreducible \(\mathfrak {so}_{2n}\)- or \(\mathfrak {sp}_{2n}\)-representations, and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian \(Y^\pm (\mathfrak {gl}_{2n})\). We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a \(\mathfrak {so}_{2n}\)- or \(\mathfrak {sp}_{2n}\)-symmetric open spin chain to that of a \(\mathfrak {gl}_{n}\)-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations.

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Acknowledgements

The authors thank Samuel Belliard, Nicolas Crampé, Nicolas Guay, Bart Vlaar and Curtis Wendlandt for useful discussions and the anonymous referee for comments and suggestions. V.R. was in part supported by the UK EPSRC under the Grant EP/K031805/1 and by the European Social Fund, Grant Number 09.3.3-LMT-K-712-02-0017. A.G. was supported by an EPSRC PhD studentship.

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Authors and Affiliations

  1. Department of Mathematics, University of York, York, YO10 5DD, UK

    Allan Gerrard, Niall MacKay & Vidas Regelskis

  2. Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio av. 3, 10257, Vilnius, Lithuania

    Vidas Regelskis

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  1. Allan Gerrard
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  2. Niall MacKay
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  3. Vidas Regelskis
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Correspondence to Vidas Regelskis.

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Communicated by Jean-Michel Maillet.

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Gerrard, A., MacKay, N. & Regelskis, V. Nested Algebraic Bethe Ansatz for Open Spin Chains with Even Twisted Yangian Symmetry. Ann. Henri Poincaré 20, 339–392 (2019). https://doi.org/10.1007/s00023-018-0731-1

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