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.