Topological insulators with hybrid-order boundary states

We report the discovery of several classes of novel topological insulators (TIs) with hybrid-order boundary states generated from the first-order TIs with additional crystalline symmetries. Unlike the current studies on hybrid-order TIs where different-order topology arises from merging different-order TIs in various energy, these novel TIs exhibit unique properties, featuring a remarkable coexistence of first-order gapless modes and higher-order Fermi arc states, behaving as a hybrid between the first-order TIs and higher-order topological semimetals within a single bulk gap. Our findings establish a profound connection between these novel $d$-dimensional ($d\mathrm{D}$) TIs and ($d\ensuremath{-}1$)D higher-order TIs (HOTIs), which can be understood as a result of stacking $(d\ensuremath{-}1)\mathrm{D}$ HOTIs to $d\mathrm{D}$ with $d=3,4$, revealing unconventional topological phase transitions by closing the gap in certain first-order boundaries rather than the bulk. The bulk-boundary correspondence between these higher-order Fermi arcs and bulk topological invariants associated with additional crystallline symmetries is also demonstrated. We then address the conventional topological phase transitions from these novel TIs to nodal-line/nodal-surface semimetal phases, where the gapless phases host new kinds of topological responses. Meanwhile, we present the corresponding tolopogical semimetal phases by stacking these unique TIs. Finally, we discuss potential ways to realize these novel phases in synthetic and real materials, with a particular focus on the feasible implementation in optical lattices using ultracold atoms.