  Microwave impedance microscopy (MIM) has been utilized to directly visualize
topological edge states in many quantum materials, from quantum Hall systems to
topological insulators, across the GHz regime. While the microwave response for
conventional metals and insulators can be accurately quantified using simple
lumped-element circuits, the applicability of these classical models to more
exotic quantum systems remains limited. In this work, we present a general
theoretical framework of the MIM response of arbitrary quantum materials within
linear response theory. As a special case, we model the microwave response of
topological edge states in a Chern insulator and predict an enhanced MIM
response at the crystal boundaries due to collective edge magnetoplasmon (EMP)
excitations. The resonance frequency of these plasmonic modes should depend
quantitatively on the topological invariant of the Chern insulator state and on
the sample's circumference, which highlights their non-local, topological
nature. To benchmark our analytical predictions, we experimentally probe the
MIM response of quantum anomalous Hall edge states in a Cr-doped (Bi,Sb)2Te3
topological insulator and perform numerical simulations using a classical
formulation of the EMP modes based on this realistic tip-sample geometry, both
of which yield results consistent with our theoretical picture. We also show
how the technique of MIM can be used to quantitatively extract the topological
invariant of a Chern insulator, disentangle the signatures of topological
versus trivial edge states, and shed light on the microscopic nature of
dissipation along the crystal boundaries.
