Metal-Insulator Coexistence and Gap-Crossing Domain-Wall Modes in an Aubry-Andr\'{e} Model with Nonlocal Hopping
Nonequilibrium transport remains a central theme in modern physics, spanning from condensed matter to synthetic systems. Here, we investigate particle transport in an extended Aubry-Andr\'e model with system-scale hopping, namely nonlocal hopping with a range proportional to the system size, and uncover a metal-insulator coexistence regime in real space, where metallic and insulating spatial domains coexist within the same system and are separated by sharp spatial boundaries. In the insulating region, particles exhibit flat-band-like localization in the absence of quasiperiodic potentials, while a quasiperiodic potential induces distinct multi-point localization, different from conventional exponential localization. Meanwhile, particles can freely propagate and tunnel across spatially disconnected metallic domains separated by the insulating region. Beyond this coexistence phase, we identify unconventional gap-crossing domain-wall modes with comb-like spatial profiles that mediate nonlocal, multi-point transport across separated metallic domains. Our findings reveal a rich interplay between localization, nonlocality, and transport in systems with nonlocal hopping.