Dense loops, supersymmetry, and Goldstone phases in two dimensions.

Loop models in two dimensions can be related to O(N) models. The low-temperature dense-loops phase of such a model, or of its reformulation using a supergroup as symmetry, can have a Goldstone broken-symmetry phase for N<2. We argue that this phase is generic for -2<N<2 when crossings of loops are allowed, and distinct from the model of noncrossing dense loops first studied by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]]. Our arguments are supported by our numerical results, and by a lattice model solved exactly by Martins et al. [Phys. Rev. Lett. 81, 504 (1998)]].