Special transitions in an O(n) loop model with an Ising-like constraint.

Abstract

We investigate the O(n) nonintersecting loop model on the square lattice under the constraint that the loops consist of 90-deg bends only. The model is governed by the loop weight n, a weight x for each vertex of the lattice visited once by a loop, and a weight z for each vertex visited twice by a loop. We explore the (x,z) phase diagram for some values of n. For 0<n<1, the diagram has the same topology as the generic O(n) phase diagram with n<2, with a first-order line when z starts to dominate and an O(n)-like transition when x starts to dominate. Both lines meet in an exactly solved higher critical point. For n>1, the O(n)-like transition line appears to be absent. Thus, for z=0, the (n,x) phase diagram displays a line of phase transitions for n≤1. The line ends at n=1 in an infinite-order transition. We determine the conformal anomaly and the critical exponents along this line. These results agree accurately with a recent proposal for the universal classification of this type of model, at least in most of the range -1≤n≤1. We also determine the exponent describing crossover to the generic O(n) universality class, by introducing topological defects associated with the introduction of "straight" vertices violating the 90-deg-bend rule. These results are obtained by means of transfer-matrix calculations and finite-size scaling.