The open XXX spin chain in the SoV framework: scalar product of separate states

Abstract

We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T-Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.

39 pages

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• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
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• On Complex Gamma-Function Integrals
• New Compact Construction of Eigenstates for Supersymmetric Spin Chains
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II
• On Scalar Products in Higher Rank Quantum Separation of Variables
• On quantum separation of variables beyond fundamental representations
• Complete spectrum of quantum integrable lattice models associated to ${\mathcal{U}}_q(\widehat{g{l}_n})$ by separation of variables
• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• On Separation of Variables for Reflection Algebras
• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
• Scalar Products in Twisted XXX Spin Chain. Determinant Representation
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• Correlation functions by Separation of Variables: the XXX spin chain
• All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Separation of variables for higher rank integrable models: review of recent progress
• On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint
• New Separation of Variables for the Classical XXX and XXZ Heisenberg Spin Chains
• Quantum Integrability and Quantum Groups: a special issue in memory of Petr P. Kulish