Dual energy-differentiated topological transition in artificial red phosphorus chains
We investigate the spectral and transport properties of an atomic chain of red phosphorus. We reveal the separation of flat-band states from the rest of the system and calculate its energy bands analytically. The topological properties of the system are established through the evaluation of the Berry (Zak) phase of the energy bands, revealing nontrivial topology. The Berry phase depends on the relative strength of the hopping parameters and exhibits dual energy-dependent topological phase transitions. Remarkably, the emergence of inert band edges provides a direct spectral signature of these transitions, acting as energy-resolved indicators of the redistribution of topological charge between bands. The existence of the associated edge states is proved numerically for finite lattices. The theoretical predictions, particularly the band structure and the existence of edge states, are further confirmed by numerical simulations of red phosphorus through a phononic lattice in the form of a highly structured aluminum plate.