An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model

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 $2^r$ 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 ${\tau }_2$-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 ${\tau }_2$-model commutes with the ${\mathfrak{sl}}_2$ loop algebra, $L({\mathfrak{sl}}_2)$, and every regular Bethe state of the ${\tau }_2$-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 ${\tau }_2$-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.

29 pages

References in corpus (2)

• Irreducibility criterion for a finite-dimensional highest weight representation of the sl(2) loop algebra and the dimensions of reducible representations
• Extension of a Borel subalgebra symmetry into the sl(2) loop algebra symmetry for the twisted XXZ spin chain at roots of unity and the Onsager algebra

Cited by in corpus (1)

• Algebraic Bethe ansatz for U(1) Invariant Integrable Models: Compact and non-Compact Applications