Observing half-integer topological winding numbers in non-Hermitian synthetic lattices.

Non-Hermitian (NH) systems have revealed unique topological phenomena that are not observed in Hermitian counterparts, such as novel topology classifications and the NH skin effect. In periodic NH systems, eigenenergies become complex and exhibit windings in the complex plane, while eigenstate winding numbers, which are strictly integers in Hermitian systems, can take half-integer values. However, direct experimental observation of NH winding of both eigenenergies and eigenstates, especially the half-integer winding, remains a significant challenge. In this work, we utilize the orbital angular momentum (OAM) synthetic dimension to construct an NH topological lattice and achieve direct observation of both eigenstate and eigenenergy windings. We report the first experimental observation of a half-integer eigenstate winding number, and reveal the intrinsic relationship between the direction in the NH skin dynamics and eigenenergy windings. Furthermore, by partitioning the OAM chain into two semi-infinite chains, we observe zero boundary modes and demonstrate that their distributions are jointly determined by the winding numbers of both the eigenstates and eigenenergies. This work provides comprehensive insights into NH topologies and offers a new experimental platform for exploring NH phenomena.