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

Introduction

Recent advancements in non-Hermitian (NH) systems have unveiled new phenomena in topological matter, which lack counterparts in Hermitian systems^{1, 2, 3, 4, 5, 6, 7–8}. For example, the number of turns that the eigenstate wavefunctions “wind” over the Brillouin zone manifold is a quantized topological invariant that governs the behavior of edge states. This principle, known as bulk-boundary correspondence, is a fundamental concept in the study of topological phases of matter. In NH system, such eigenstate winding can exhibit half-integer values^9,10, deviating from the integer winding numbers typical of Hermitian systems. Additionally, the eigenenergies become complex and exhibit windings in the complex plane¹¹ under periodic boundary conditions, which signal the presence of the NH skin effect¹².

The half-integer eigenstate windings, together with the complex eigenenergy windings, play a crucial role in enhancing the understanding of topological properties in NH systems and in elucidating the NH bulk-boundary correspondence. However, the NH skin effect introduces extreme sensitivity to boundary conditions, leading to substantial modifications in the spectrum and the failure of conventional bulk-boundary correspondence. Under open boundary conditions, the NH bulk-boundary correspondence is appropriately described by the non-Bloch winding number^{13, 14, 15–16}. In alternative configurations, such as semi-infinite chains that avoid two-end coupling, the characterization of edge states necessitates the use of non-Hermitian half-integer winding numbers^10,17,18.

To date, eigenenergy windings in NH systems have been extensively studied across various platforms, including photonic systems^11,19,20, acoustic metamaterials²¹, ion traps²², and mechanical membranes^7,23. Additionally, half-integer charges in the eigenstates around exceptional points have been experimentally observed⁴. However, the eigenstate windings over the Brillouin zone manifold, which are directly associated with edge states, have only been observed in closed Hermitian systems^24,25, where the winding numbers are strictly integer-valued. In this work, we leverage synthetic dimension^{26, 27, 28–29} based on the orbital angular momentum (OAM)^30,31 to investigate the nontrivial topological windings of both eigenenergies and eigenstates in an NH topological lattice. We not only establish the intrinsic connection between eigenenergy windings and NH skin dynamics³², but also report, for the first time, the observation of a half-integer winding number as the eigenstates wind over...