Measurement protocol for detecting correlated topological insulators in synthetic quantum systems

Two-dimensional topological insulators, characterized by symmetry-protected anomalous boundary modes, have been generalized to the strongly correlated regime for both bosonic and fermionic systems. As correlated topological insulators (TI) approach experimental realization in quantum simulators, conventional probes, such as transport measurements, are not easily applicable to these synthetic platforms. In this study, we focus on two examples of correlated TI: a bosonic TI protected by $\mathbb{Z}_2\times U(1)$ symmetry and the fermionic quantum spin Hall insulator protected by time-reversal symmetry. We propose a unified, readily implementable protocol based on measuring the disorder parameter $\langle\exp(\mathrm{i}\theta \hat{Q}_A)\rangle$ for a large subregion $A$, with $\hat{Q}_A$ the total charge operator within $A$. Our key finding is that this quantity exhibits non-analytical dependence on $\theta$ for correlated TI, a signature robust against decoherence. We establish this diagnostic through both numerical simulations and analytical derivations. This protocol is well-suited for implementation on near-term quantum simulation platforms, providing a direct route to experimentally confirm correlated TI.