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


We investigate the O(nn) 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 nn, a weight xx for each vertex of the lattice visited once by a loop, and a weight zz for each vertex visited twice by a loop. We explore the (x,z)(x,z) phase diagram for some values of nn. For 0<n<10<n<1, the diagram has the same topology as the generic O(nn) phase diagram with n<2n<2, with a first-order line when zz starts to dominate, and an O(nn)-like transition when xx starts to dominate. Both lines meet in an exactly solved higher critical point. For n>1n>1, the O(nn)-like transition line appears to be absent. Thus, for z=0z=0, the (n,x)(n,x) phase diagram displays a line of phase transitions for n1n\le 1. The line ends at n=1n=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 1n1-1 \leq n \leq 1. We also determine the exponent describing crossover to the generic O(nn) 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.Comment: 19 pages, 11 figure

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