Probing Local Topology in a Disordered Higher-Order Topological Insulator

Higher-order topology is prized for its ability to realize lower-dimensional boundary states which are stable beyond fine-tuning. However, disorder presents a failure mechanism that can destroy topological in-gap states. Here, we investigate a disordered two-dimensional polariton lattice and employ the spectral localizer framework to define a real-space topological index rooted in crystalline spatial symmetries. This framework enables direct real-space mapping of topology beyond conventional momentum-space classifications, confirming the presence of corner and edge modes in this generalized Su-Schrieffer-Heeger model. Furthermore, it can directly quantify topological protection of a state. We leverage the versatility of our platform to experimentally realize normally distributed, random disorder and find that the corner states persist until the spectral gap closes. Experimentally, this corresponds to a disorder strength of approximately one quarter of the spectral gap. The spectral localizer accurately identifies the disorder strength at which the bandgap closes, establishing the framework as a predictive tool for every finite size system. Our results broaden the design principles for higher-order topological insulators and open the way towards imple menting disorder-resilient devices for robust lasing, light-routing, and quantum computation.