$Z_2$ topological signature of the optical bound on maximal Berry curvature: Application to two-dimensional time-reversal symmetric insulators

Unlike broken time-reversal symmetric (TRS) systems with a well-defined Chern number, directly measuring the bulk $Z_2$ invariant and Berry curvature (if nonzero) in topological insulators and their higher-order topological families remains an unsolved problem. Here, based on the refined trace-determinant inequality (TDI) involving the trace and determinant of the quantum metric and maximal Berry curvature (MBC), we propose an optical bound on the MBC for two-dimensional TRS insulators. As a result, using experimental data from a series of measurements, where the band-inversion parameter is tuned and the optical conductivity is measured over a certain energy range, one can identify the $Z_2$ topological signature and construct the topological phase diagram by integrating the optical bound over frequency. This is supported by the momentum integration of the refined TDI, $f$-sum rule, and its topological extension, which provide a topological lower bound. To clearly identify the $Z_2$ topological signature, the faster decay of the optical weight in the topologically trivial region is crucial; this faster decay can be controlled by the optical gap and the inverse mass tensor. Crucially, the MBC we introduced enables us to prove the existence of tight topological lower bounds for the optical weight, quantum weight, and the double quantum volume. Based on the tight topological lower bounds, their physical meaning can be interpreted as an upper bound on the number of boundary states. We illustrate our approach using three representative topological models: the Kane-Mele model, mirror-protected insulator, and quadrupole insulator. Our results demonstrate that the MBC can reveal symmetry protected-topology and plays a role analogous to that of the original Berry curvature.