Linking Numbers for Self-Avoiding Loops and Percolation: Application to the Spin Quantum Hall Transition

Nonlocal twist operators are introduced for the $\mathrm{O}\left(n\right)$ and $Q$-state Potts models in two dimensions which count the numbers of self-avoiding loops (respectively, percolation clusters) surrounding a given point. Their scaling dimensions are computed exactly. This yields many results: for example, the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. Its mean behaves as $\left(1/3\sqrt{3}\pi \right)|\ln (p_c-p)|$ as $p\to p_c-$. As an application we compute the exact value $\sqrt{3}/2$ for the conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.