SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra

Abstract

The analysis of the transfer matrices associated to the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method which generalizes to these integrable quantum models the method first introduced by Sklyanin. More in detail, for the representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which apply in particular to transfer matrix eigenstates. The comparison of our analysis for the 8-vertex reflection algebra with that of [1, 2] for the 6-vertex one leads to the interesting remark that a profound similarity in both the characterization of the spectral problems and of the scalar products exists for these two different realizations of the reflection algebra once they are described by SOV method. As it will be shown in a future publication, this remarkable similarity will be at the basis of the simultaneous determination of form factors of local operators of integrable quantum models associated to general reflection algebra representations of both 8-vertex and 6-vertex type.

37 pages

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Cited by in corpus (22)

• The open XXX spin chain in the SoV framework: scalar product of separate states
• On quantum separation of variables
• Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I
• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• On determinant representations of scalar products and form factors in the SoV approach: the XXX case
• Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• 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
• Correlation functions by Separation of Variables: the XXX spin chain
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions
• Invariant subspaces and explicit Bethe vectors in the integrable open spin $1/2$ $\XYZ$ chain
• 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
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint