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An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model

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Abstract

In terms of the \(\mathfrak{sl}_{2}\) loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of 2r eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the τ 2-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the τ 2-model commutes with the \(\mathfrak{sl}_{2}\) loop algebra, \(L(\mathfrak{sl}_{2})\) , and every regular Bethe state of the τ 2-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the τ 2-model generates an \(L(\mathfrak{sl}_{2})\) -degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.

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Authors and Affiliations

  1. Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan

    Akinori Nishino

  2. Department of Physics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610, Japan

    Tetsuo Deguchi

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  1. Akinori Nishino
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  2. Tetsuo Deguchi
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Correspondence to Akinori Nishino.

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Nishino, A., Deguchi, T. An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model. J Stat Phys 133, 587–615 (2008). https://doi.org/10.1007/s10955-008-9624-x

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  • DOI: https://doi.org/10.1007/s10955-008-9624-x

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