Localized states coupled to a network of chiral modes in minimally twisted bilayer graphene

Minimally twisted bilayer graphene in the presence of an interlayer bias develops a triangular network of valley chiral modes that propagate along the $AB\mathrm{/}BA$ interfaces and scatter at the $AA$ regions. The low energy physics of the resulting network can be captured by means of a phenomenological scattering network model, allowing to calculate the energy spectrum and the magnetoconductance in a straightforward way. Although there is in general a good agreement between microscopic and phenomenological models, there are some aspects that have not been captured so far with the latter. In particular, the appearance of flatbands in the energy spectrum associated to a localized density of states at the $AA$ regions. To bring both approaches closer together, we modify the previous energy independent phenomenological model and add the possibility to scatter to a set of discrete energy levels at the $AA$ regions, yielding a $S$ matrix with energy dependent parameters. Furthermore, we investigate the impact of Coulomb repulsion in these regions on a mean-field level and discuss possible effects of decoherence due to elastic and inelastic cotunneling events.