In Hermitian systems, single-particle topological invariants are quantized to integers, like the winding number and Chern number1. Nevertheless, non-Hermitian systems2,3,4, which incorporate energy exchange with external environments, can exhibit fractional topological invariants like half-integers. Theoretical studies have predicted that in one-dimensional non-Hermitian lattices, two distinct winding numbers can be defined: one based on eigenvectors (denoted as w, akin to the Zak phase)3,5,6 and another based on complex energy (denoted as v)7. The combinations of these two types of winding numbers govern the presence of edge states under open-boundary conditions. For example, v = 0 and w = 0 yielding no edge states, v = 0 and w = 1 producing two edge states at both ends (Fig. 1a), and |v | = 1 with w = 1/2 resulting in a single edge state in a semi-infinite chain (Fig. 1b), highlighting their unique behavior4,5. So far, the experimental demonstration of these predictions has been lacking.

Fig. 1: Non-Hermitian lattice model via the synthetic orbital angular momentum dimension and bulk-boundary correspondence.
figure 1

a The topological edge states at both ends of the non-Hermitian OAM lattice chain when v = 0 and w = 1. b A single edge state at the single end of a semi-infinite OAM lattice chain when v = −1 and w = 1/2. c Schematic figure of a degenerate optical cavity composed of a Q-plate, a wave plate, and a partially polarized beam splitter. The Q-plate generates intracell hoppings among distinct lattice sites by partially transferring spin to orbital (i.e., m) of resonant modes. The wave plate induces intercell hoppings among OAM modes, and the partially polarized beam splitter introduces non-Hermiticity. By drilling a proper hole at the center of both the wave plate and the beam splitter, two semi-infinite lattice chains are created, in which a single edge state jointly determined by v and w (i.e., panel b) is observable

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A recent study by Yang et al.8, realized non-Hermitian edge modes jointly controlled by the eigenstate and eigenvalue winding numbers. The team successfully created a one-dimensional non-Hermitian lattice model using synthetic dimensions9,10 constructed from orbital angular momenta (OAMs) in an optical cavity11,12 (Fig. 1c). By leveraging the OAM of light as lattice sites, the researchers introduced non-Hermiticity via a partially polarized beam splitter and provided the experimental evidence of this half-integer non-Hermitian topological invariant w = 1/2. Unlike prior experiments observing half-integer winding numbers in the vicinity of exceptional points, the half-integer valued eigenstate winding number observed in this study is defined across the full Brillouin zone. Importantly, a semi-infinite chain was realized, verifying a single edge state of the half-integer eigenstate winding number.

To create a domain wall, the team partitioned the infinite OAM chain into two semi-infinite chains by strategically drilling a proper hole in the wave plate and partially polarized beam splitter, blocking the propagation of higher-order OAM modes (Fig. 1c). This setup effectively mimicked the open-boundary condition of a semi-infinite lattice (Fig. 1b)13, enabling the observation of edge-state characteristics of such systems. The authors performed polarization-resolved transmission measurements to map out both eigenenergy and eigenstate windings. The experiment directly confirmed the theoretical prediction that for w = 1/2 and |v| = 1, a single edge state emerges at the end of the semi-infinite chain, with its position determined by the sign of v. This work represents the experimental validation of the correspondence between half-integer winding numbers and edge states in semi-infinite non-Hermitian chains, bridging a critical gap between theories and experiments.

The reported measurement incorporates both spectral and transport characteristics. Such high controllability in synthetic non-Hermitian systems may find potential applications in a broader context. The flexibility of the synthetic dimension framework, particularly using OAM modes, offers a versatile platform for exploring photonic topological phenomena, such as edge modes of semi-infinite domain wall configurations of non-Hermitian quantum walks14 and the interplay between non-Hermiticity and quantum entanglement15,16. Moreover, the polarization-resolved and quasi-momentum-resolved (relating to the azimuthal coordinate of light in this case) measurement performed here could be particularly useful for creating non-Abelian topology17 based on the OAM of light.