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

We investigate the O($n$) nonintersecting loop model on the square lattice under the constraint that the loops consist of ninety-degree 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 $n\le 1$. 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 $-1 \leq n \leq 1$. 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 ninety-degree-bend rule. These results are obtained by means of transfer-matrix calculations and finite-size scaling.

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