Topological Signatures of the Optical Bound on Maximal Berry Curvature: Applications to Two-Dimensional Time-Reversal-Symmetric Insulators

Abstract: Unlike broken time-reversal symmetric (TRS) systems with a 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 establish an optical bound on the MBC for two-dimensional TRS insulators. By utilizing experimental data on linearly polarized optical conductivity within a certain energy range, the topological signatures can be identified by integrating the optical bound. We illustrate our approach using three representative topological models: the Kane-Mele model, mirror-protected insulator, and quadrupole insulator. We find that the resulting momentum integration of the refined TDI provides an intrinsic relationship between quantum volume (QV) variations and topological phase transitions. We argue that double QV can be treated as an upper bound on the number of boundary states. Our findings offer a method for extracting the topological signatures of TRS insulators using optical conductivity data.