Fig. 2: Schematic diagrams of SSH model and corresponding band structures and wavefunction distributions.

a, b Schematic of gain-induced semi-infinite SSH model (a), and infinite SSH model (b) supporting topological edge modes. In (a), the boundary is designated as the gain at leftmost lattice site together with its coupled neighbour site, while the bulk region on the right contributes with self-energy to the boundary. In (b), the interface region is designated as an arbitrary intermediate site together with its coupled neighbour site, and it accounts for self-energy from both left and right semi-infinite segments of the bulk. Red-coloured site represents the gain site. c, d The complex spectra and parity-time (PT) phase transition of topological edge modes (c) and topological interface modes (d). Red-coloured curves represent bands of edge (interface) modes, blue-dotted curves and purple-dotted curves are projection of edge (interface) modes on real-energy plane and imaginary-energy plane, respectively, and blue-shaded regions are bulk bands. e, f Wavefunction profile variations over the lattice as a function of the gain parameter Γ, for the schematic shown in (a) and (b), respectively. The purple and blue arrows indicate the location of edge states emergence and PT phase transition. Coupling parameters v = 0.95 and w = 0.208 are used consistently throughout the manuscript unless otherwise specified.