Fig. 3

Theoretical analysis of \(p\)-orbital HOTIs: topological invariant and corner-mode robustness. a Bulk polarization of the upper three bands (blue, green, and purple as plotted in (b1) and (b2) as a function of the orbital coupling ratio \({t}_{\pi }/{t}_{\sigma }\), exhibiting step jumps at the band-crossing points (marked by vertical dashed lines). The sum of the quantized polarizations of the upper three bands is \(\left(0+1/3+2/3\right) mod 1=0\). b1–b2 Band structures plotted for (b1) \({t}_{\pi }/{t}_{\sigma }=-0.19\) and (b2) \({t}_{\pi }/{t}_{\sigma }=-0.77\) at \({t}_{1}/{t}_{2}=\text{0.6 and }\theta =4\pi /3\), showing the change in band-crossing for the upper three bands. c Band structure of the auxiliary Hamiltonian \({H}^{{\prime}}({\varvec{k}},\theta )\) calculated for \({t}_{1}/{t}_{2}=\text{0.6}\) and \({t}_{\pi }/{t}_{\sigma }=-0.8\) at \(\theta ={0}\), which is equivalent to two sets of decoupled \(s\)-band BKLs: red for one set and dark blue for the other set. d 3D plot of the band structure of \({H}^{{\prime}}({\varvec{k}},\theta )\) as a function of \(\theta\), showing that the gap at zero energy remains open for any \(\theta\) (or for that matter, any orbital hybridization). e Calculated winding numbers, where the white dotted line marks the orbital-hopping-symmetry condition \({t}_{1\sigma }/{t}_{2\sigma }={t}_{1\pi }/{t}_{2\pi }\) required for topological protection of the orbital corner states. Here, circles in the lower inset illustrate distinct windings for each case, and the blue cross in the \(\mathcal{W}=2\) region corresponds to the experimental parameters used to realize the \(p\)-orbital HOTI. f1–f2 Robustness test of orbital corner states in a rhombic BKL obtained by applying random perturbations with increasing strengths \(\delta\) between site couplings that (f1) preserve the A-SubSy (i.e., without A-A coupling) and (f2) break the A-SubSy (i.e., with A-A coupling). The corner modes (red circles) remain at zero energy in (f1) but not in (f2). See Additional file 1 for details