A random matrix approach to topological invariants: The winding number
A talk in the Seminar Zufallsmatrizen series by
Nico Hahn from University of Duisburg-Essen
| Abstract: | Topological non-triviality of disordered quantum matter manifests itself in localized states at
the boundary of the solid body. The amount of these topological edge states is saved in the
topological invariant, whose concrete mathematical nature depends on the symmetries of
the system. Generally, for disordered quantum systems, we distinguish between the ten
Altland-Zirnbauer symmetry classes, that rely on time reversal invariance, particle-hole
conjugation and chiral symmetry. In our work we considered the chiral unitary class AIII and
the chiral symplectic class CII in one dimension. Here, the topological invariant is related to
the concept of the winding number in complex analysis. We set up a parametric random
matrix model for the chiral Hamiltonians, making the Winding number random. Our goal is
to obtain the correlation functions of the winding numbers, which we expect to be universal
in an unfolding limit. We trace this problem back to an average over ratios of characteristic
polynomials, involving the spherical ensemble of matrices $K_1^{-1} K_2$, where $K_1$, $K_2$ are
Ginibre distributed. We tackle this problem by employing a technique that exhibits
reminiscent supersymmetric structures, while we never carry out any map to superspace. Please contact Mario Kieburg (m.kieburg@unimelb.edu.au) for details regarding access. Within the CRC this talk is associated to the project(s): C6 |