PT Unknown
AU Hahn M. Sc., N
TI Statistical Topology - The Winding Number in One-Dimensional Chiral Systems
PD 03
PY 2024
DI 10.17185/duepublico/81773
LA en
DE Theoretical Physics; Mathematical Physics; Random Matrix Theory; Topology
AB Topological insulators and topological superconductors are characterized by edge states that evade disorder-induced localization. The number of these states is determined by a topological invariant. Whether a topologically non-trivial phase is possible depends on the symmetries and the dimension of the system. The topological invariant relevant for one-dimensional systems with chiral symmetry is the winding number. In this work, we perform a statistical analysis of the winding number in the framework of random matrix theory. Random matrix theory is known to produce universal results for systems with a sufficient degree of complexity in the limit of large matrix dimensions. In the context of solid state physics, this complexity corresponds to disorder, i.e. spatially inhomogeneous perturbations of the system parameters. In order to conduct our study, we first set up a parametric random matrix model with chiral symmetry for the Bloch Hamiltonian. In addition to chiral symmetry, we also classify our model based on the presence or absence of time reversal invariance. Specifically, we calculate the correlations of the winding number density, which yield the statistical moments of the winding number upon integration, as well as the distribution of the winding number. On a technical level, we trace the topological problem back to a spectral one, which renders the toolbox of random matrix theory applicable. In doing so, we encounter the spherical ensemble of random matrices, which, unlike the classical ensembles of random matrix theory, does not follow a Gaussian matrix probability distribution. We employ different methods of random matrix theory to carry out the ensemble averages. In particular, we work with a technique that is related to the supersymmetry method of random matrix theory. It exploits supersymmetry structures without reformulating the problem in superspace and is therefore also referred to as supersymmetry without supersymmetry.
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