Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker

Abstract

In 2D semiconductors and insulators, the Chern number of the valence band Bloch state is an important quantity that has been linked to various material properties, such as the topological order. We elaborate that the opacity of 2D materials to circularly polarized light over a wide range of frequencies, measured in units of the fine structure constant, can be used to extract a spectral function that frequency-integrates to the Chern number, offering a simple optical experiment to measure it. This method is subsequently generalized to finite temperature and locally on every lattice site by a linear response theory, which helps to extract the Chern marker that maps the Chern number to lattice sites. The long range response in our theory corresponds to a Chern correlator that acts like the internal fluctuation of the Chern marker, and is found to be enhanced in the topologically nontrivial phase. Finally, from the Fourier transform of the valence band Berry curvature, a nonlocal Chern marker is further introduced, whose decay length diverges at topological phase transitions and therefore serves as a faithful indicator of the transitions, and moreover can be interpreted as a Wannier state correlation function. The concepts discussed in this work explore multi-faceted aspects of topology and should help address the impact of system inhomogeneities.

Berry curvature and quantum metric in copper-substituted lead phosphate apatite

“…Finally, we remark that in semiconductors and insulators, one can define a Berry curvature spectral function [28] Ω xy (k, ω) that frequency-integrates to the Berry curvature Ω xy (k) = ´dω Ω xy (k, ω) and momentum-integrates to the Chern number spectral function C(ω) = ´d2 k (2π) 2 Ω xy (k, ω), with the later being related to the circular dichroism of 2D materials [35]. Likewisely, one can also introduce a quantum metric spectral function [28] g µν (k, ω) that frequencyintegrates to the quantum metric g µµ (k) = ´dω g µµ (k, ω) and is proportional the optical conductivity g µµ (k, ω) ∝ σ µµ (k, ω)/ω at momentum k, which can possibly be detected in pump-probe type of experiments [27].…”