The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

Abstract

The spectral properties of operators formed from generators of the q-Onsager non-Abelian infinite dimensional algebra are investigated. Using a suitable functional representation, all eigenfunctions are shown to obey a second-order q-difference equation (or its degenerate discrete version). In the algebraic sector associated with polynomial eigenfunctions (or their discrete analogues), Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz approach is not applicable in related massive quantum integrable models, the eigenfunctions are also described. The spin-half XXZ open spin chain with general integrable boundary conditions is reconsidered in light of this approach: all the eigenstates are constructed. In the algebraic sector which corresponds to special relations among the parameters, known results are recovered.

arXiv comment17 pages; LaTeX file with amssymb; v2: typos corrected, references added, to appear in Nucl.Phys.B

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Cited by in corpus (51)

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• The Universal Askey-Wilson Algebra
• The open XXX spin chain in the SoV framework: scalar product of separate states
• Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields
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• Complete Bethe Ansatz solution of the open spin-s XXZ chain with general integrable boundary terms
• Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms
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• A classification of sharp tridiagonal pairs
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• 3-state Hamiltonians associated to solvable 33-vertex models
• Generalized T-Q relations and the open spin-s XXZ chain with nondiagonal boundary terms
• Exact ground state and elementary excitations of a competing spin chain with twisted boundary condition
• Off-diagonal Bethe Ansatz for the $D_3^{(1)}$ model
• Exact solution of the $sp(4)$ integrable spin chain with generic boundaries
• Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices
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• Asymmetric Twin Representation: the Transfer Matrix Symmetry
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• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Generic boundary scattering in the open XXZ chain
• Bethe States of the integrable spin-s chain with generic open boundaries
• Spectrum of the transfer matrices of the spin chains associated with the $A_3^{(2)}$ Lie algebra
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• Double Lowering Operators on Polynomial
• Graded Off-diagonal Bethe ansatz solution of the $SU(2|2)$ spin chain model with generic integrable boundaries