Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators

Abstract

The integrable quantum models, associated to the transfer matrices of the 6-vertex reflection algebra for spin 1/2 representations, are studied in this paper. In the framework of Sklyanin's quantum separation of variables (SOV), we provide the complete characterization of the eigenvalues and eigenstates of the transfer matrix and the proof of the simplicity of the transfer matrix spectrum. Moreover, we use these integrable quantum models as further key examples for which to develop a method in the SOV framework to compute matrix elements of local operators. This method has been introduced first in [1] and then used also in [2], it is based on the resolution of the quantum inverse problem (i.e. the reconstruction of all local operators in terms of the quantum separate variables) plus the computation of the action of separate covectors on separate vectors. In particular, for these integrable quantum models, which in the homogeneous limit reproduce the open spin-1/2 XXZ quantum chains with non-diagonal boundary conditions, we have obtained the SOV-reconstructions for a class of quasi-local operators and determinant formulae for the covector-vector actions. As consequence of these findings we provide one determinant formulae for the matrix elements of this class of reconstructed quasi-local operators on transfer matrix eigenstates.

41 pages, minor misprints and one remark after Lemma 5.1 added

References in corpus (49)

• Separation of Variables. New Trends.
• How Algebraic Bethe Ansatz works for integrable model
• Form factors of the XXZ Heisenberg spin-1/2 finite chain
• Reaction-Diffusion Processes, Critical Dynamics and Quantum Chains
• Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries
• Computation of dynamical correlation functions of Heisenberg chains in a field
• On the quantum inverse scattering problem
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• XXZ chain with a boundary
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Functional relations from the Yang-Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions
• Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms
• Difference Equations in Spin Chains with a Boundary
• Separation of Variables in the open XXX chain
• Q-operator and T-Q relation from the fusion hierarchy
• Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions
• The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach
• A deformed analogue of Onsager's symmetry in the XXZ open spin chain
• Separation of variables for integrable spin-boson models
• Hidden Grassmann structure in the XXZ model V: sine-Gordon model
• Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions
• One-boundary Temperley-Lieb algebras in the XXZ and loop models
• Null-vectors in Integrable Field Theory
• Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra
• Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models
• Bethe Ansatz for the Temperley-Lieb loop model with open boundaries
• Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity
• On the second reference state and complete eigenstates of the open XXZ chain
• Counting the local fields in SG theory.
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• Bethe Ansatz derived from the functional relations of the open XXZ chain for new special cases
• Correlation functions of the higher spin XXX chains
• The space of local operators in perturbed conformal field theories,
• The XXZ model with anti-periodic twisted boundary conditions
• Structure of Matrix Elements in Quantum Toda Chain
• The Sine-Gordon model revisited I
• Generalized coordinate Bethe ansatz for non diagonal boundaries
• Form factors of integrable Heisenberg (higher) spin chains
• Spin Chains with Non-Diagonal Boundaries and Trigonometric SOS Model with Reflecting End
• Bethe Ansatz solution of the open XX spin chain with nondiagonal boundary terms
• Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end
• On the algebraic Bethe Ansatz approach to the correlation functions of the XXZ spin-1/2 Heisenberg chain
• Counting minimal form factors of the restricted sine-Gordon model
• Isomorphism of critical and off-critical operator spaces in two-dimensional quantum field theory
• Fermionic screening operators in the sine-Gordon model
• Fermionic basis for space of operators in the XXZ model
• On domain wall boundary conditions for the XXZ spin Hamiltonian
• Bethe Ansatz for the open XXZ chain from functional relations at roots of unity
• New results in the XXZ open spin chain

Cited by in corpus (76)

• Off-diagonal Bethe ansatz and exact solution of a topological spin ring
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields
• The half-infinite XXZ chain in Onsager's approach
• Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions
• Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz
• Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
• Modified algebraic Bethe ansatz for XXZ chain on the segment - II - general cases
• Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms
• Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof
• From the Quantum Transfer Matrix to the Quench Action: The Loschmidt echo in $XXZ$ Heisenberg spin chains
• Integrable approach to simple exclusion processes with boundaries. Review and progress
• The open XXX spin chain in the SoV framework: scalar product of separate states
• Algebraic Bethe ansatz for open XXX model with triangular boundary matrices
• 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
• Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields
• 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
• Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries
• Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms
• SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra
• On the boundaries of quantum integrability for the spin-1/2 Richardson-Gaudin system
• Correlation functions of the half-infinite XXZ spin chain with a triangular boundary
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• Phantom Bethe roots in the integrable open spin $1/2$ $XXZ$ chain
• Scale-free non-Hermitian skin effect in a boundary-dissipated spin chain
• Integrable quenches in nested spin chains II: fusion of boundary transfer matrices
• Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary
• Algebraic Bethe ansatz for the six vertex model with upper triangular $K$-matrices
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II
• On Scalar Products in Higher Rank Quantum Separation of Variables
• Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries
• Chiral coordinate Bethe ansatz for phantom eigenstates in the open XXZ spin-$\frac{1}{2}$ chain
• 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
• Exact boundary free energy of the open XXZ chain with arbitrary boundary conditions
• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
• On Separation of Variables for Reflection Algebras
• Slavnov and Gaudin-Korepin formulas for models without $U(1)$ symmetry: the XXX chain on the segment
• 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
• Form factors of the half-infinite XXZ spin chain with a triangular boundary
• Inversion identities for inhomogeneous face models
• Truncation identities for the small polaron fusion hierarchy
• Non-Abelian $SU(3{)}_k$ anyons: inversion identities for higher rank face models
• Aspects of the inverse problem for the Toda chain
• Bethe Ansatz solution of the small polaron with nondiagonal boundary terms
• All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint
• 3-state Hamiltonians associated to solvable 33-vertex models
• Spectrum of the quantum integrable $D_2^{(2)}$ spin chain with generic boundary fields
• Off-diagonal Bethe Ansatz for the $D_3^{(1)}$ model
• Thermodynamic limit and boundary energy of the spin-1 Heisenberg chain with non-diagonal boundary fields
• Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables
• Exact surface energy of the $D_2^{(1)}$ spin chain with generic non-diagonal boundary reflections
• A representation basis for the quantum integrable spin chain associated with the su(3) algebra
• The solution of an open XXZ chain with arbitrary spin revisited
• Exact solution of the quantum integrable model associated with the twisted $D_3^{(2)}$ algebra
• On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model
• Inhomogeneous discrete-time exclusion processes
• Time and band limiting operator and Bethe ansatz
• A convenient basis for the Izergin-Korepin model
• 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
• Bethe States of the integrable spin-s chain with generic open boundaries
• Spectrum of the transfer matrices of the spin chains associated with the $A_3^{(2)}$ Lie algebra
• Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint
• Fermionic reflection matrices
• A note on a boundary sine-Gordon model at the free-Fermion point