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

TL;DR

This paper introduces an optical bound on the maximal Berry curvature in two-dimensional time-reversal symmetric insulators, enabling topological signatures to be identified through optical conductivity measurements.

Contribution

It establishes a novel optical bound based on the refined trace-determinant inequality, linking optical data to topological invariants in TRS insulators.

Findings

Optical bounds can identify topological signatures in insulators.

Quantized quantum volume relates to the Gauss-Bonnet theorem.

Double quantum volume bounds boundary state count.

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 the optical conductivity within a certain energy range, the topological signatures can be identified through the frequency integration of the optical bound. This is supported by the momentum integration of the refined TDI and its $f$-sum rule and topological extension, which provide a topological lower bound. Meanwhile, the decay of optical weight in the topologically trivial region can be controlled by the optical gap and the inverse mass tensor. We illustrate our approach using three representative topological models: the Kane-Mele model, mirror-protected insulator, and quadrupole insulator. Remarkably, we find that the quantized quantum volume (QV) in the mirror-protected insulator results from the Gauss-Bonnet theorem. Since QV has a topological lower bound, this suggests that double QV can be interpreted 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.