Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I

Abstract

We study the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. The results apply as well to the spectral analysis of the lattice sine-Gordon model with integrable open boundary conditions. This spectral analysis is developed by implementing the method of separation of variables (SoV). The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions. Moreover, we prove an equivalent characterization as the set of solutions to a Baxter's like T-Q functional equation and rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form. In order to explain our method in a simple case, the present paper is restricted to representations containing one constraint on the boundary parameters and on the parameters of the Bazhanov-Stroganov Lax operator. In a next article, some more technical tools (like Baxter's gauge transformations) will be introduced to extend our approach to general integrable boundary conditions.

arXiv comment39 pages, minor misprints corrected, some references added

References in corpus (94)

• Many-Body Physics with Ultracold Gases
• Quantum many-body systems out of equilibrium
• Collapse and Revival of the Matter Wave Field of a Bose-Einstein Condensate
• Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas
• Boundary S-Matrix and Boundary State in Two-Dimensional Integrable Quantum Field Theory
• Non-equilibrium coherence dynamics in one-dimensional Bose gases
• Open XXZ spin chain: Nonequilibrium steady state and strict bound on ballistic transport
• Separation of Variables. New Trends.
• Microscopic observation of magnon bound states and their dynamics
• Fermionic transport in a homogeneous Hubbard model: Out-of-equilibrium dynamics with ultracold atoms
• Form factors of the XXZ Heisenberg spin-1/2 finite chain
• Reaction-Diffusion Processes, Critical Dynamics and Quantum Chains
• Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensions
• Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries
• Diffusion and ballistic transport in one-dimensional quantum systems
• Boundary K-Matrices for the Six Vertex and the n(2n-1) A_{n-1} Vertex Models
• On the quantum inverse scattering problem
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Boundary K-matrices for the XYZ, XXZ AND XXX spin chains
• Algebraic Structures Related to Reflection Equations
• Exact Spectral Gaps of the Asymmetric Exclusion Process with Open Boundaries
• Functional relations and Bethe Ansatz for the XXZ chain
• Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles
• Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory
• XXZ chain with a boundary
• 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 Quantum Inverse Scattering Method for Hubbard-like Models
• Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity
• 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
• The half-infinite XXZ chain in Onsager's approach
• Correlation functions of the open XXZ chain I
• Difference Equations in Spin Chains with a Boundary
• Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators
• Separation of Variables in the open XXX chain
• Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz
• Q-operator and T-Q relation from the fusion hierarchy
• Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
• Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions
• 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
• Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain
• Quantization of Solitons and the Restricted Sine-Gordon Model
• Correlation functions of the open XXZ chain II
• Separation of variables for the quantum SL(2,R) spin chain
• A deformed analogue of Onsager's symmetry in the XXZ open spin chain
• Separation of variables for integrable spin-boson models
• Algebraic Bethe ansatz for eight vertex model with general open-boundary conditions
• Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions
• Null-vectors in Integrable Field Theory
• Transfer matrix functional relations for the generalized tau_2(t_q) model
• The open XXX spin chain in the SoV framework: scalar product of separate states
• Integrable spin-boson models descending from rational six-vertex models
• Integrable Boundary Conditions for the One-Dimensional Hubbard Model
• Algebraic Bethe ansatz for open XXX model with triangular boundary matrices
• Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models
• 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
• 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
• Fused integrable lattice models with quantum impurities and open boundaries
• Spectrum and Bethe ansatz equations for the U_ {q}(gl(N)) closed and open spin chains in any representation
• Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV
• Equivalences between spin models induced by defects
• Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables
• The XXZ model with anti-periodic twisted boundary conditions
• Cyclic monodromy matrices for sl(n) trigonometric R-matrices
• Algebraic Bethe ansatz for the one-dimensional Hubbard model with open boundaries
• Structure of Matrix Elements in Quantum Toda Chain
• Baxter-Bazhanov-Stroganov model: Separation of Variables and Baxter Equation
• Analytical Bethe Ansatz for closed and open gl(M|N) super-spin chains in arbitrary representations and for any Dynkin diagram
• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• Analytical Bethe Ansatz for closed and open gl(n)-spin chains in any representation
• The Sine-Gordon model revisited I
• Generalized coordinate Bethe ansatz for non diagonal boundaries
• Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors
• On determinant representations of scalar products and form factors in the SoV approach: the XXX case
• Complete Bethe Ansatz solution of the open spin-s XXZ chain with general integrable boundary terms
• Form-factors in the Baxter-Bazhanov-Stroganov model I: Norms and matrix elements
• Nested Bethe ansatz for `all' open spin chains with diagonal boundary conditions
• SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra
• The open XXZ and associated models at q root of unity
• Form-factors in the Baxter-Bazhanov-Stroganov model II: Ising model on the finite lattice
• The 8-vertex model with quasi-periodic boundary conditions
• The Highest Weight Property for the $S{U}_q(n)$ Invariant Spin Chains
• Eigenvectors in the Superintegrable Model I: sl_2 Generators
• On the form factors of local operators in the lattice sine-Gordon model
• Non-Abelian symmetries of the half-infinite XXZ spin chain
• The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method
• Eigenvectors in the Superintegrable Model II: Ground State Sector
• Exact solution of the lattice vertex model analog of the coupled Bariev XY chains
• Factorized finite-size Ising model spin matrix elements from Separation of Variables

Cited by in corpus (17)

• The open XXX spin chain in the SoV framework: scalar product of separate states
• On quantum separation of variables
• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
• 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
• Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model
• 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
• On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint