Separation of variables for integrable spin-boson models

Abstract

We formulate the functional Bethe ansatz for bosonic (infinite dimensional) representations of the Yang-Baxter algebra. The main deviation from the standard approach consists in a half infinite 'Sklyanin lattice' made of the eigenvalues of the operator zeros of the Bethe annihilation operator. By a separation of variables, functional TQ equations are obtained for this half infinite lattice. They provide valuable information about the spectrum of a given Hamiltonian model. We apply this procedure to integrable spin-boson models subject to both twisted and open boundary conditions. In the case of general twisted and certain open boundary conditions polynomial solutions to these TQ equations are found and we compute the spectrum of both the full transfer matrix and its quasi-classical limit. For generic open boundaries we present a two-parameter family of Bethe equations, derived from TQ equations that are compatible with polynomial solutions for Q. A connection of these parameters to the boundary fields is still missing.

30 pages, revtex4. Minor changes

References in corpus (7)

• Many-Body Physics with Ultracold Gases
• Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Functional relations from the Yang-Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions
• Separation of Variables in the open XXX chain
• Integrable spin-boson models descending from rational six-vertex models
• Quasi-classical descendants of disordered vertex models with boundaries

Cited by in corpus (31)

• On the Integrability of the Rabi Model
• Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators
• Integrability vs exact solvability in the quantum Rabi and Dicke models
• The open XXX spin chain in the SoV framework: scalar product of separate states
• Solution of the Dicke model for N=3
• Functional Bethe ansatz methods for the open XXX chain
• On quantum separation of variables
• Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables
• 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
• SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• Algebraic Bethe Ansätze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models
• 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
• A large class of solvable multistate Landau-Zener models and quantum integrability
• 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
• Domain wall partition function of the eight-vertex model with a non-diagonal reflecting end
• Application of polaron picture in the two-qubit quantum Rabi model
• Scalar products of the open XYZ chain with non-diagonal boundary terms
• 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
• Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end
• Quasi-Exactly Solvable Models Derived from the Quasi-Gaudin Algebra
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Drinfeld twist and symmetric Bethe vectors of the open XYZ chain with non-diagonal boundary terms
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint