Linking numbers for self-avoiding loops and percolation: Application to the spin quantum Hall transition

Nonlocal twist operators are introduced for the O(n) and Q-state Potts mode ls in two dimensions which count the numbers of self-avoiding loops (respec tively, percolation clusters) surrounding a given point. Their scaling dime nsions are computed exactly. This yields many results: for example, the num ber of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. Its mean behaves as (1/3 root 3 pi)\ ln( p(c) - p)\ as p --> p(c)-. As an application we compute the exact value roo t 3/2 for the conductivity at the spin Hall transition, as well as the shap e dependence of the mean conductance in an arbitrary simply connected geome try with two extended edge contacts.