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

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 twoand three-state vertex models. These models include the six-vertex model, the 15-vertex A (1) 2 model and the 19-vertex models of Izergin-Korepin and ZamolodchikovFateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomogeneities. 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 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. Physics and Astronomy Classifications. 05.50.+q