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

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Cited by in corpus (17)

• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• Spin operator matrix elements in the quantum Ising chain: fermion approach
• The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method
• Eigenvectors in the Superintegrable Model II: Ground State Sector
• Serre-Lusztig relations for $ı$quantum groups
• Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra
• Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ
• Spin operator matrix elements in the superintegrable chiral Potts quantum chain
• Identities in the Superintegrable Chiral Potts Model
• Algebraic Bethe ansatz for U(1) Invariant Integrable Models: Compact and non-Compact Applications
• Some ground-state expectation values for the free parafermion Z(N) spin chain
• Spontaneous Magnetization of the Integrable Chiral Potts Model
• Quantum Loop Subalgebra and Eigenvectors of the Superintegrable Chiral Potts Transfer Matrices
• Analogues of Lusztig's higher order relations for the q-Onsager algebra
• Reduction formula of form factors for the integrable spin-s XXZ chains and application to the correlation functions
• Serre Relations in the Superintegrable Model
• Correlation functions for the three state superintegrable chiral Potts spin chain of finite lengths