Linking numbers for self-avoiding walks and percolation: application to the spin quantum Hall transition

TL;DR

This paper introduces non-local twist operators in two-dimensional models to analyze linking numbers for self-avoiding walks and percolation, leading to exact results for the spin quantum Hall transition.

Contribution

It develops a conformal field theory approach with twist operators to compute linking numbers and applies this to derive exact physical quantities at the spin Hall transition.

Findings

Exact value of cb2/2 for the conductivity at the spin Hall transition

Distribution of percolation clusters crossing a point

Shape dependence of mean conductance in arbitrary geometries

Abstract

Non-local twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which, in the limits n -> 0 (resp. Q -> 1) count the numbers of self-avoiding loops (resp. percolation clusters) surrounding a given point. This yields many results, for example the distribution of the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. These twist operators correspond to (1,2) in the Kac classification of conformal field theory, so that their higher-point correlations, which describe linking numbers around multiple points, may be computed exactly. As an application we compute the exact value \sqrt 3/2 for the dimensionless 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.