Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices

Abstract

The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the six-vertex model, the 15-vertex $A_2^{(1)}$ model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomeogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices {\em do not} coincide with those which lead to quantum group-invariant spin chains. The exact solution of the integrable loop models -- including an $O(n)$ model on the square lattice with open boundaries -- is of relevance to the surface critical behaviour of two-dimensional polymers.

35 pages, LaTeX with PostScript figures; minor corrections, version to appear in Nucl. Phys. B

References in corpus (10)

• Boundary S-Matrix and Boundary State in Two-Dimensional Integrable Quantum Field Theory
• Boundary K-Matrices for the Six Vertex and the n(2n-1) A_{n-1} Vertex Models
• Addendum to ``Integrability of Open Spin Chains with Quantum Algebra Symmetry''
• Analytical Bethe Ansatz for Quantum-Algebra-Invariant Spin Chains
• Exact solution of the $S{U}_q(n)$ invariant quantum spin chains
• Solvable lattice models labelled by Dynkin diagrams
• Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice
• Exact Bethe Ansatz solution for $A_{n-1}$ chains with non-$S{U}_q(n)$ invariant open boundary conditions
• On the construction of integrable dilute ADE models
• The Field Theory Limit of Integrable Lattice Models

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• Row Transfer Matrix Functional Relations for Baxter's Eight-Vertex and Six-Vertex Models with Open Boundaries Via More General Reflection Matrices
• Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions
• Algebraic Bethe ansatz for eight vertex model with general open-boundary conditions
• Matrix-Product States for a One-Dimensional Lattice Gas with Parallel Dynamics
• Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice
• Solutions of the reflection equation for face and vertex models associated with $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$
• Drude weight in systems with open boundary conditions
• Duality and quantum-algebra symmetry of the A_{N-1}^(1) open spin chain with diagonal boundary fields
• Analytical Bethe Ansatz for $A_{2n-1}^{(2)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ quantum-algebra-invariant open spin chains
• Non compact conformal field theory and the a_2^{(2)} (Izergin-Korepin) model in regime III
• Conformal boundary conditions in the critical O(n) model and dilute loop models
• Complete Bethe Ansatz solution of the open spin-s XXZ chain with general integrable boundary terms
• On $A_{n-1}^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)},A_{2n}^{(2)},A_{2n-1}^{(2)}$ and $D_{n+1}^{(2)}$ Reflection $K$-Matrices
• Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms
• Bethe Ansatz solution for quantum spin-1 chains with boundary terms
• Integrability of the $D_n^2$ vertex models with open boundary
• Bethe Ansatze for 19-vertex Models
• Finite lattice Bethe ansatz systems and the Heun equation
• Exact solution of the anisotropic special transition in the O(n) model in 2D
• Integrability and conformal data of the dimer model
• Algebraic Bethe Ansatz for the Zamolodchikov-Fateev and Izergin-Korepin models with open boundary conditions
• Surface Critical Phenomena and Scaling in the Eight-Vertex Model
• Lattice regularisation of a non-compact boundary conformal field theory
• Integrable O(n) model on the honeycomb lattice via reflection matrices : Surface critical behaviour
• The principal indecomposable modules of the dilute Temperley-Lieb algebra
• Static properties of 2D spin-ice as a sixteen-vertex model
• An Ising model in a magnetic field with a boundary
• Integrable supersymmetric correlated electron chain with open boundaries
• Nonstandard coproducts and the Izergin-Korepin open spin chain
• An open-boundary integrable model of three coupled XY spin chains
• A unified treatment of Ising model magnetizations
• Heisenberg XYZ Hamiltonian with Integrable Impurities
• Cylinder partition function of the 6-vertex model from algebraic geometry
• Exact solution and surface critical behaviour of open cyclic SOS lattice models
• Algebraic Bethe Ansatz solutions for the $sl(2|1{)}^{(2)}$ and $osp(2|1)$ models with boundary terms
• Classical topological paramagnetism
• Edge states and conformal boundary conditions in super spin chains and super sigma models
• Surface Critical Phenomena in Interaction-Round-a-Face Models
• Fusion hierarchies, $T$-systems and $Y$-systems for the $A_2^{(1)}$ models
• Surface operator content of the A_L face models
• Extraordinary transition in the two-dimensional O(n) model
• Quantum group symmetries and completeness for A_{2n}^(2) open spin chains
• Critical Behaviour of Spanning Forests on Random Planar Graphs
• The algebraic Bethe ansatz for open A_{2n}^{(2)} vertex model
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry
• Integrability of the Heisenberg Chains with Boundary Impurities and Their Bethe Ansatz
• Reflection $K$-matrices for a nineteen vertex model with $U_q[\mathrm{osp}{\left(2|2\right)}^{\left(2\right)}]$ symmetry
• Algebraic Bethe ansatz for 19-vertex models with upper triangular K-matrices
• Integrable eight-state supersymmetric $U$ model with boundary terms and its Bethe ansatz solution
• A convenient basis for the Izergin-Korepin model
• The O(n) loop model on the 3-12 lattice
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice
• Fusion and monodromy in the Temperley-Lieb category
• Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip
• Exact surface energies and boundary excitations of the Izergin-Korepin model with generic boundary fields
• The spectrum of an open vertex model based on the U_q[SU(2)] at roots of unity