Solutions of the Boundary Yang-Baxter Equation for A–D–E Models

A two-dimensional lattice spin model in statistical mechanics can be considered as solvable with periodic boundary conditions if its bulk Boltzmann weights satisfy the Yang-Baxter equation [1], and as additionally solvable with non-periodic boundary conditions if it admits boundary weights which satisfy the boundary Yang-Baxter equation [2]. Many such models are now known. Restricting our attention to interaction-round-a-face models, these are the eight-vertex solid-on-solid model [3], the cyclic solid-on-solid models [4], the AL models [5, 6, 7], the fused AL models [6], the dilute AL models [8] and certain higher rank models associated with A n , B (1) n , C (1) n , D (1) n and A (2) n [9]. Here, we present general forms of the boundary weights for some of these previously-considered models, and for some additional, related models. We begin, in Section 2, by stating the standard relations, including the Yang-Baxter equation and the boundary Yang-Baxter equation, which may be satisfied by the bulk and boundary weights of an interaction-round-a-face model, and we define two important types of boundary weight, diagonal and non-diagonal. In Section 3, we consider certain intertwiner properties which may be satisfied by the bulk and boundary weights of two appropriatelyrelated interaction-round-a-face models. In Sections 4–9, we obtain boundary weights, mostly of the diagonal type, which represent general solutions of the boundary Yang-Baxter equation for various models, including the standard and dilute AL, DL and E6,7,8 models. We conclude, in Section 10, with a discussion of general techniques for solving the boundary Yang-Baxter equation.