Interplay between long-range hopping and disorder in topological systems

We extend the standard Su-Schrieffer-Heeger (SSH) model to include long-range hopping amplitudes and disorder, and analyze how the electronic and topological properties are affected. We show that long-range hopping can change the symmetry class and the topological invariant, while diagonal and off-diagonal disorder lead to Anderson localization. Interestingly, we find that the Lyapunov exponent $\ensuremath{\gamma}(E)$ can be linked in two ways to the topological properties in the presence of disorder---either due to the different response of midgap states to increasing disorder or due to an extra contribution to $\ensuremath{\gamma}$ due to the presence of edge modes. Finally, we discuss its implications in realistic transport measurements.