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Twisted bilayer graphene gives rise to large moir\'{e} patterns that form atriangular network upon mechanical relaxation. If gating is included, eachtriangular region has gapped electronic Dirac points that behave as bulktopological insulators with topological indices depending on valley index andthe type of stacking. Since each triangle has two oppositely charged valleys,they remain topologically trivial.  In this work, we address several questions related to the edge currents ofthis system by analysis and computation of continuum PDE models. Firstly, wederive the bulk invariants corresponding to a single valley, and then apply abulk-interface correspondence to quantify asymmetric transport along theinterface. Secondly, we introduce a valley-coupled continuum model to show howvalleys are approximately decoupled in the presence of small perturbationsusing a multiscale expansion, and how valleys couple for larger defects.Thirdly, we present a method to prove for a large class of continuum(pseudo-)differential models that a quantized asymmetric current is preservedthrough a junction such as a triangular network vertex. We support all of thesearguments with numerical simulations using spectral methods to compute relevantcurrents and wavepacket propagation.

Mathematical models of topologically protected transport in twisted bilayer graphene