Dense Loops, Supersymmetry, and Goldstone Phases in Two Dimensions

Loop models in two dimensions can be related to $\mathrm{O}(N)$ 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)].

Figure 1

Measures of the central charge vs $w$ for $N=0$. The conjectured value for $L\to \infty$ is $c=-1$ for all $w>0$.

Figure 2

The $k$-leg exponents $X_k=2h_k$ vs $w$ for $N=0$.

Figure 3

Measures of $c-24h_{\mathrm{t}\mathrm{w}}$ vs $w$ for $N=0$. Dashed lines show the prediction of Eq. (4).