Statistical Topology: A Random Matrix Approach to the Winding Number

Topological insulators and topological superconductors are characterized
by edge states that are stable against perturbations. The number of edge
states is determined by a topological invariant defined for the bulk
system. The origin of this effect lies in the interplay between symmetry
and topology. In disordered systems there are only ten symmetry classes,
called Altland-Zirnbauer classes, which are based on three fundamental
symmetries: time reversal invariance, particle-hole invariance and
chiral symmetry.

In our work we consider one-dimensional systems with chiral symmetry.
The relevant topological invariant for this class of systems is the
winding number. We set up a random matrix model for the Bloch
Hamiltonian describing a disordered unit cell. Random matrix theory is a
versatile tool and often capable of modelling universal statistical
properties in complex systems. Within this model, we compute the
statistical moments and the distribution of the winding number.