Dilute oriented loop models

Abstract

We study a model of dilute oriented loops on the square lattice, where each loop is compatible with a fixed, alternating orientation of the lattice edges. This implies that loop strands are not allowed to go straight at vertices, and results in an enhancement of the usual symmetry to . The corresponding transfer matrix acts on a number of representations (standard modules) that grows exponentially with the system size. We derive their dimension and those of the centralizer by both combinatorial and algebraic techniques. A mapping onto a field theory permits us to identify the conformal field theory governing the critical range, . We establish the phase diagram and the critical exponents of low-energy excitations. For generic n, there is a critical line in the universality class of the dilute model, terminating in an point. The case n = 1 maps onto the critical line of the six-vertex model, along which exponents vary continuously.