Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof

Abstract

In this paper, we prove the off-shell equation satisfied by the transfer matrix associated with the XXZ spin-$\frac{1}{2}$ chain on the segment with two generic integrable boundaries acting on the Bethe vector. The essential step is to prove that the expression of the action of a modified creation operator on the Bethe vector has an off-shell structure which results in an inhomogeneous term in the eigenvalues and Bethe equations of the corresponding transfer matrix.

V2 published version, 16 pages

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• Why scalar products in the algebraic Bethe ansatz have determinant representation
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• The open XXZ spin chain in the SoV framework: scalar product of separate states
• Scale-free non-Hermitian skin effect in a boundary-dissipated spin chain
• Modified Algebraic Bethe Ansatz: Twisted XXX Case
• Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary
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• Bethe ansatz for an AdS/CFT open spin chain with non-diagonal boundaries
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• On Separation of Variables for Reflection Algebras
• Scalar Products in Twisted XXX Spin Chain. Determinant Representation
• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
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• A note on ${\mathfrak{gl}}_2$-invariant Bethe vectors
• Slavnov and Gaudin-Korepin formulas for models without $U(1)$ symmetry: the XXX chain on the segment
• Off-diagonal Bethe Ansatz on the $so(5)$ spin chain
• Correlation functions by Separation of Variables: the XXX spin chain
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• Scalar product for the XXZ spin chain with general integrable boundaries
• Exact solution of an su(n) spin torus
• All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint
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• 3-state Hamiltonians associated to solvable 33-vertex models
• 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
• Bethe ansatz diagonalization of the Heun-Racah operator
• Algebraic Bethe ansatz for the Temperley-Lieb spin-1 chain
• A representation basis for the quantum integrable spin chain associated with the su(3) algebra
• Exact solution of the quantum integrable model associated with the twisted $D_3^{(2)}$ algebra
• T-W relation and free energy of the Heisenberg chain at a finite temperature
• Bethe states of the trigonometric SU(3) spin chain with generic open boundaries
• Computation of entanglement entropy in inhomogeneous free fermions chains by algebraic Bethe ansatz
• Surface energy and elementary excitations of the XYZ spin chain with integrable open boundary fields
• Time and band limiting operator and Bethe ansatz
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Elementary excitations in an integrable twisted J1-J2 spin chain in the thermodynamic limit
• Exact solution of the trigonometric SU(3) spin chain with generic off-diagonal boundary reflections
• On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model
• Spectrum of the transfer matrices of the spin chains associated with the $A_3^{(2)}$ Lie algebra
• Bethe States of the integrable spin-s chain with generic open boundaries
• Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields
• "Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models
• On the ${\mathcal{U}}_q[osp(1|2)]$ Temperley-Lieb model
• Graded Off-diagonal Bethe ansatz solution of the $SU(2|2)$ spin chain model with generic integrable boundaries