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.

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

References in corpus (4)

• Bethe Ansatz Equations of XXZ Model and q-Sturm-Liouville Problems
• Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations
• Aspects of the ODE/IM correspondence
• An Atavistic Lie Algebra

Cited by in corpus (23)

• The Universal Askey-Wilson Algebra
• Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields
• Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms
• Exact ground state and elementary excitations of a topological spin chain
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint
• Tridiagonal pairs of $q$-Racah type
• Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials
• Exact ground state and elementary excitations of a competing spin chain with twisted boundary condition
• Asymmetric Twin Representation: the Transfer Matrix Symmetry
• The solution of an open XXZ chain with arbitrary spin revisited
• Root patterns and exact surface energy of the spin-1 Heisenberg model with generic open boundaries
• Off-diagonal approach to the exact solution of quantum integrable systems
• Elementary excitations in an integrable twisted J1-J2 spin chain in the thermodynamic limit
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• On the shape of a tridiagonal pair
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Surface energy and elementary excitations of the XYZ spin chain with integrable open boundary fields
• Spectrum of the transfer matrices of the spin chains associated with the $A_3^{(2)}$ Lie algebra
• Exact Lorentz-violating $q$-deformed O($N$) universality class
• Graded Off-diagonal Bethe ansatz solution of the $SU(2|2)$ spin chain model with generic integrable boundaries
• Double Lowering Operators on Polynomial