Fig. 2: Demonstration of the A-SubSy-protected topological state in SSH lattice.
a, The SSH model. b, Spectra of the unperturbed (black circles) and the A-SubSy-perturbed system (t1 = 0.1, t2 = 1). In all subplots, the blue crosses are perturbed bands, the red circle is the perturbed left edge state at zero energy, and the green cross is the perturbed right edge state. c, Modal structure of the unperturbed (black) and the perturbed left (red) and right (green) edge states. The black modes are shifted up for better visibility. d,e, Left: experimental images of A-SubSy-preserving lattice with B–B coupling (d) and A-SubSy-breaking lattice with A–A coupling (e). The white arrows indicate that solely the left-most waveguide (A sublattice site) is excited at z = 0. Middle: output intensity patterns of the probe beam after propagating 20 mm through corresponding lattices in the left panels, respectively. The intensity profile in the A-SubSy-preserving lattice resides solely on the A sublattice (d), whereas the intensity profile in the A-SubSy-breaking lattice is present on both A and B sublattices (e), which demonstrates SubSy protection of the left topological edge mode. The lattice constant (size of the unit cell) is 40 μm, and the twist angle is 57°. Right: numerical simulations in the middle panels agree with the related measurements. f,g, Left: experimental images of A-SubSy-preserving zigzag lattice with periodic B–B coupling (f) and A-SubSy-breaking lattice with A–A coupling (g). Middle: output intensity patterns after 20 mm of propagation (the probe beam excites the bottom site). The intensity profile in the A-SubSy-preserving lattice resides solely on the A sublattice (f), whereas it leaks onto the B sublattice for the A-SubSy-breaking case (g). This again demonstrates the SubSy protection of the edge mode \(\left| {{A_{\mathrm{L}}}} \right\rangle\). For these experiments, the lattice constant is 55 μm, and the twist angle is 60°. Right: numerical simulations to ten-times-longer propagation distance corroborate our experimental results.
