Emergent dual topology in the three-dimensional Kane-Mele Pt₂HgSe₃

Recently, the very first large-gap Kane-Mele quantum spin Hall insulator was predicted to be monolayer jacutingaite (Pt₂HgSe₃), a naturally-occurring exfoliable mineral. The stacking of quantum spin Hall monolayers typically leads to a (0;001) weak topological phase, which does not protect the existence of surface states on the (001) surface. Unexpectedly, recent ARPES experiments revealed the presence of surface states dispersing over large areas of the 001-surface Brillouin zone. Such 001-surface states have been shown to be topologically protected by a non-zero mirror Chern number, associated with a nodal line gapped by spin-orbit interactions. Here, we extend the two-dimensional Kane-Mele model to bulk jacutingaite and unveil the microscopic origin of the gapped nodal line and the emerging crystalline topological order. By using maximally-localized Wannier functions, we identify a large non-trivial second nearest-layer hopping term that breaks the standard paradigm of weak topological insulators. Complemented by this term, the predictions of the Kane-Mele model are in remarkable agreement with recent experiments and first-principles simulations, providing an appealing conceptual framework also relevant for other layered materials made of stacked honeycomb lattices.