An Effective Theory for Graphene Nanoribbons with Junctions

Graphene nanoribbons are a promising candidate for fault-tolerant quantum electronics. In this scenario, qubits are realised by localised states that can emerge on junctions in hybrid ribbons formed by two armchair nanoribbons of different widths. We derive an effective theory based on a tight-binding ansatz for the description of hybrid nanoribbons and use it to make accurate predictions of the energy gap and nature of the localisation in various hybrid nanoribbon geometries. We use quantum Monte Carlo simulations to demonstrate that the effective theory remains applicable in the presence of Hubbard interactions. We discover, in addition to the well known localisations on junctions, which we call `Fuji', a new type of `Kilimanjaro' localisation smeared out over a segment of the hybrid ribbon. We show that Fuji localisations in hybrids of width $N$ and $N+2$ armchair nanoribbons occur around symmetric junctions if and only if $N\pmod3=1$, while edge-aligned junctions never support strong localisation. This behaviour cannot be explained relying purely on the topological $Z_2$ invariant, which has been believed the origin of the localisations to date.