Fig. 1

Illustration of “zero-energy” \(p\)-orbital topological corner states in a BKL. a Schematic diagram of the triangular BKL with six-unit cells, where the white dashed triangle marks one unit cell consisting of three sublattices (A, B, and C), and \({t}_{1}\) and \({t}_{2}\) indicate the intracell and intercell hopping amplitudes, respectively. Each lattice site corresponds to a laser-written waveguide, which supports \(p\)-orbital modes (see the left inset). Two types of orbital hopping amplitudes, \({t}_{\sigma }\) and \({t}_{\pi }\), are illustrated in the right inset. In the top vertex of the lattice, we depict the rotation of an orbital corner state under nonlinear excitation. b Calculated eigenvalue spectrum \({\beta }_{L}\) as a function of \({t}_{1}/{t}_{2}\) at \({t}_{\pi }/{t}_{\sigma }=-0.3\) for the finite-size BKL shown in (a), where distinct “zero-energy” corner states in the highly nontrivial regime are marked in red. c Calculated band structure for the BKL with \({t}_{1}/{t}_{2}=0.1\), showing six corner states (red dots) at \({\beta }_{L}\approx 0\), where two representatives (\({p}_{x}\)- and \({p}_{y}\)-type modes relative to the top corner) are plotted in (d1, d2). Note that a topological \(p\)-orbital corner mode, as the \({p}_{x}\)-mode illustrated in (d1), is characteristically localized at the corner site 1, with a staggered phase distribution in the NNN site 3 but zero amplitude at the nearest-neighbor site 2