3D Loop Models and the ${\mathrm{CP}}^{n-1}$ Sigma Model

Many statistical mechanics problems can be framed in terms of random curves; we consider a class of three-dimensional loop models that are prototypes for such ensembles. The models show transitions between phases with infinite loops and short-loop phases. We map them to ${\mathrm{CP}}^{n-1}$ sigma models, where $n$ is the loop fugacity. Using Monte Carlo simulations, we find continuous transitions for $n=1$, 2, 3, and first order transitions for $n\ge 5$. The results are relevant to line defects in random media, as well as to Anderson localization and ($2+1$)-dimensional quantum magnets.

Figure 1

Pairings at a node (with associated weights), and the labeling of links used in (2).

Figure 2

Loops on the 3D $K$ lattice at $p=0$. At $p=1$, they become infinite straight trajectories (crossing at nodes).

Figure 3

Phase diagrams for the $K$ and $L$ lattices. Continuous transitions are indicated by blue dots and single line and first order transitions by red dots and double line.

Figure 4

Scaling collapse for $n_w$ ($p$, $L$) on the 3D $K$ lattice. Main panel∶ $n=2$; inset∶ $n=3$.

Figure 5

First order transitions at $n=5$∶ (main panel) the jump in $n_w$ on the $K$ lattice and (inset) in $n_{+}$ on the 3D $L$ lattice.