Layer dependent topological phases and transitions in TaRhTe4 : From monolayer and bilayer to bulk

The recently synthesized ternary quasi-2D material ${\mathrm{TaRhTe}}_{4}$ is a bulk Weyl semimetal with an intrinsically layered structure, which poses the question of how the topology of its electronic structure depends on layer separations. Experimentally, these separations may be changed for instance by intercalation of the bulk, or by exfoliation, to reach monolayer or few-layer structures. Here, we show that in the monolayer limit a quantum spin Hall insulator (QSHI) state emerges, employing density functional calculations as well as a minimal four-orbital tight-binding model that we develop. Even for weak spin-orbit couplings the QSHI is present, which has an interesting edge state that features Rashba-split bands with quadratic band minima. Further, we find that a weak topological insulator (WTI) manifests in the bilayer system due to sizable intralayer hopping, contrary to the common lore that only weak interlayer interactions between stacked QSHIs lead to WTIs. Stacked bilayers give rise to a phase diagram as a function of the interlayer separation that comprises a Weyl semimetal, WTI, and normal insulator (NI) phases. These insights on the evolution of topology with dimensions can be transferred to the family of layered ternary transition metal tellurides.