Quantum transport in graphene-based Chern mosaics

Chern mosaics arise due to the changing local Chern number between different domains in the bulk of the system. This change leads to the emergence of chiral modes along domain walls that are protected from backscattering, making them interesting for transport purposes. The important topological invariant in graphene-based Chern mosaics is the local valley Chern number, which changes oppositely in the two inequivalent valleys in the Brillouin zone. Thus, the boundary modes propagate in opposite directions in the different valleys. As long as these valleys are well-separated in reciprocal space, backscattering is prohibited under the assumption that any disorder present in real space is sufficiently smooth. The goal of this thesis is to describe these Chern mosaics in terms of scattering networks based on the Chalker-Coddington network. We analyze transport characteristics, such as network bands for a translationally invariant system and conductance through a finite-size strip connected to metallic leads. We incorporate a finite local density of states, represented by interacting discrete energy levels, into the established scattering network descriptions of minimally twisted bilayer graphene with an interlayer bias, which has been disregarded in previous publications. The scattering matrix calculated using the generalized Mahaux-Weidenmüller formula exhibits an energy-dependence. We show that this model can be tuned into a high and low conducting regime, depending on the momentum difference between the two modes in the same link. Furthermore, we observe different regimes depending on the energy of the modes relative to the localized states. We construct a phenomenological Kagome network based on the Chern mosaic arising in e.g. hBN-graphene-hBN-heterostructures and show how this network can be mapped to a triangular network, finding an energy-dependent scattering matrix due to the dynamical phase accumulated on the contracted links. We perform spectrum calculations and compare these to the conductance of a finite network strip. Furthermore, we analyze the effect of a non-Abelian phase that arises from a finite Rashba spin-orbit coupling and causes a localization effect. We calculate a scattering network with finite superconducting pairing at the scattering regions. Such a system may arise from superconducting correlations within twisted bilayer graphene or from proximitizing the Chern mosaic of minimally twisted bilayer graphene with a superconductor. The resulting network bands show gaps in the single-particle spectrum. The superconducting pairing couples electrons and holes in different valleys, which move in different directions. Thus, these modes accumulate the same Peierls phase in the presence of a finite magnetic field, resulting in Aharonov-Bohm oscillations even without a finite particle deflection probability.