Marginality of bulk-edge correspondence for single-valley Hamiltonians

We study the correspondence between the non-trivial topological properties associated with the individual valleys of gapped bilayer graphene (BLG), as a prototypical multi-valley system, and the gapless modes at its edges and other interfaces. We find that the exact connection between the valley-specific Hall conductivity and the number of gapless edge modes does not hold in general, but is dependent on the boundary conditions, even in the absence of intervalley coupling. This non-universality is attributed to the absence of a well-defined topological invariant within a given valley of BLG; yet, a more general topological invariant may be defined in certain cases, which explains the distinction between the BLG-vacuum and BLG-BLG interfaces.