Gapless edge states localized to odd/even layers of AA′-stacked honeycomb multilayers with staggered AB-sublattice potentials

In honeycomb multilayers with staggered AB-sublattice potentials, we predict gapless edge states localized to either of the odd and the even layers for the AA ^′ stacking order in which the sublattice-pseudospin polarizations of adjacent layers are antiparallel. Gaps in the projected layer-pseudospin spectrum suppress interlayer hopping between odd and even layers. The layer-valley Chern number corresponding to the edge states was obtained by decomposing the occupied state into two layer-pseudospin sectors by using a projected layer-pseudospin operator. For the AB ^′ stacking, the sublattice-pseudospin polarizations of adjacent layers are antiparallel, but the layer-pseudospin spectrum gap closes at the interface of the topologically different states, leading to gapped edge states. For the AA and AB stackings where the sublattice-pseudospin polarizations of the adjacent layers are parallel, the gapless edge states corresponding to quantum valley Hall states are evenly distributed across the layers.