Geometric percolation at the surface of a topological superconductor

We present evidence that strongly suggests the equivalence between disordered surface states of topological superconductors (TSCs) and geometric percolation. Percolation is known to play a role in quantum Hall systems with magnetic fields. Our unexpected result implies that percolation applies to TSC surface states, in the absence of time-reversal symmetry breaking. Moreover, the usual "even-odd" effect that occurs in such a system (as identified by Pruisken in the integer quantum Hall effect and by Haldane for spin chains) is found to be absent.

We numerically study the surface states of time-reversal TSCs in class CI with generic bulk winding numbers and quenched disorder. The low-energy states are predicted to be described by a Wess-Zumino-Witten conformal field theory (WZW-CFT), with universal wave function statistics that depend only on the bulk winding number. In this class, finite energy surface states were expected to be Anderson localized. We verify WZW-CFT results at low-energy, but find critical delocalization at finite energy. The statistics at finite energy are universal and match those of the spin quantum Hall plateau transition in class C, which is equivalent to percolation. Our result suggests that quantum Hall effects in 2D are closely linked to TSCs in 3D.