Fig. 1: Tight-binding models (TBMs).

a, b Sketch of the TBM for phases I and II, respectively. c Top: \({\theta }_{z}\) evolution of the projected dispersion for a supercell of phase I that is periodic in the \(y\) direction but has \(15\) unit cells in the \(x\) direction, simulated at \({k}_{y}={{{{{\rm{\pi }}}}}}\). Bottom: The same as top panel, but for a supercell of phase I that is periodic in the \(x\) direction but has \(15\) unit cells in the \(y\) direction, simulated at \({k}_{x}={{{{{\rm{\pi }}}}}}\). d \({\theta }_{z}\) evolution of the projected dispersion for a 2D finite system of phase I that has \(15\) unit cells in both \(x\) and \(y\) directions, where gapless corner states (red and magenta dots, black and green stars) appear in the common band gap of the bulk and edge states. In c and d, the cyan (blue) dots are the twofold degenerate dots, indicating the left and right (top and bottom) edge states, and the olive dots represent the bulk states. e Typical eigenfields of the corner states in d. f, g are the same as c and d, respectively, but for phase II. h 2D finite system constructed by placing phases I and II together. i The same as d, but for the structure shown in h. j 2D structure constructed by cutting the structure shown in h along the red dashed line. k The same as i, but for the structure in j.