Observation of spinful square-root and higher-order topological phases enriched by synthetic spin–orbit coupling

Topological phases induced by spin–orbit coupling (SOC) have been widely investigated and realized in various artificial systems. In this work, we theoretically propose and experimentally demonstrate two distinct spinful topological phases enriched by one- and two-dimensional SOC using topological circuit metamaterials. First, we report the emergence of a spinful square-root topological phase in a one-dimensional SOC-induced Aharonov–Bohm (AB) caging lattice model, characterized by spin-dependent topological winding numbers. Second, we present a spinful higher-order topological phase in a two-dimensional dimerized SOC lattice model, which hosts eight topological corner modes and is characterized by multipole chiral numbers. Experimentally, we synthesize effective spin–orbit couplings in a circuit metamaterial and implement both lattice models, with impedance measurements directly revealing their characteristic topological boundary and corner responses. Our work provides new physical insights into SOC-induced and SOC-enriched synthetic topological phases.