  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.
