The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint

Abstract

We consider the open XYZ spin chain with boundary fields. We solve the model by the new Separation of Variables approach introduced in arXiv:1904.00852. In this framework, the transfer matrix eigenstates are obtained as a particular sub-class of the class of so-called separate states. We consider the problem of computing scalar products of such separate states. As usual, they can be represented as determinants with rows labelled by the inhomogeneity parameters of the model. We notably focus on the special case in which the boundary parameters parametrising the two boundary fields satisfy one constraint, hence enabling for the description of part of the transfer matrix spectrum and eigenstates in terms of some elliptic polynomial Q-solution of a usual TQ-equation. In this case, we show how to transform the aforementioned determinant for the scalar product into some more convenient form for the consideration of the homogeneous and thermodynamic limits: as in the open XXX or XXZ cases, our result can be expressed as some generalisation of the so-called Slavnov determinant.

34 pages, minor changes, Appendix A added

References in corpus (21)

• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Correlation functions of the open XXZ chain I
• Separation of Variables in the open XXX chain
• Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions
• Correlation functions of the open XXZ chain II
• Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model
• Functional Bethe ansatz methods for the open XXX chain
• SOS model partition function and the elliptic weight functions
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I
• Baxter-Bazhanov-Stroganov model: Separation of Variables and Baxter Equation
• Why scalar products in the algebraic Bethe ansatz have determinant representation
• The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• Scalar product for the XXZ spin chain with general integrable boundaries
• Large-distance and long-time asymptotic behavior of the reduced density matrix in the non-linear Schrödinger model
• All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint
• Overlap between usual and modified Bethe vectors
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Scalar products of Bethe vectors in the generalized algebraic Bethe ansatz