Fig. 1: Variable-order topological insulator with phase transition between the spin-Chern insulator (SCI) and second-order topological insulator (SOTI).

a Bilayer breathing kagome lattice model with three inequivalent sites (gray circles) labeled as 1–3 in each layer of a unit cell. Blue (red) line denotes the NN intralayer hopping \({t}_{0}-{d}_{t}\) (\({t}_{0}+{d}_{t}\)). Green line denotes the chiral interlayer hopping \({t}_{b}\). b The first Brillouin zone (red rhombic) with high symmetry points. c Phase diagram in the \({t}_{b}/{t}_{0}\) versus \({d}_{t}/{t}_{0}\) plane. White lines are the phase boundaries with bulk gap closure. Normal insulator in the phase diagram is abbreviated as NI. d Bulk dispersions with \({d}_{t}/{t}_{0}=-0.3\) (point A), \({d}_{t}/{t}_{0}=0.4\) (point B), \({d}_{t}/{t}_{0}=0.8\) (point C). e Projected dispersions of a ribbon with open boundary along the \(y\) direction for parameters at point A in (c). Inset: the illustrations of the ribbon and the edge states propagations. f Eigenvalue of a triangle-shaped sample (inset) for parameters at point C.