Edge Transport in Haldane-Like Models with Quasi-Periodic Disorder

We consider two-dimensional topological insulators on the cylinder, in the presence of weak quasi-periodic disorder. In the absence of disorder, we assume the presence of a single edge state on the boundary of the system, as for the Haldane model. We prove that, at large distances, the boundary correlations agree with the correlations of a renormalized, translation-invariant, massless relativistic model in $1+1$ dimensions, multiplied by non-universal oscillatory factors, incommensurate with the lattice spacing. Furthermore, we compute the edge conductance and the edge susceptibility, starting from Kubo formula. We obtain explicit expressions for these response functions, completely determined by the renormalized Fermi velocity of the edge modes. In particular, we prove the quantization of the edge conductance, and the non-universality of the susceptibility. The proof relies on multiscale analysis and rigorous renormalization group methods for quasi-periodic systems, and on lattice Ward identities.