  Topological flat moir\'e bands with nearly ideal quantum geometry have been
identified in homobilayer transition metal dichalcogenide moir\'e
superlattices, and are thought to be crucial for understanding the fractional
Chern insulating states recently observed therein. Previous work proposed
viewing the system using an adiabatic approximation that replaces the
position-dependence of the layer spinor with a nonuniform periodic effective
magnetic field. When the local zero-point kinetic energy of this magnetic field
cancels identically against that of an effective Zeeman energy, a Bloch-band
version of Aharonov-Casher zero-energy modes, which we refer to as
Aharonov-Casher band, emerges leading to ideal quantum geometry. Here, we
critically examine the validity of the adiabatic approximation and identify the
parameter regimes under which Aharonov-Casher bands emerge. We show that the
adiabatic approximation is accurate for a wide range of parameters including
those realized in experiments. Furthermore, we show that while the cancellation
leading to the emergence of Aharonov-Casher bands is generally not possible
beyond the leading Fourier harmonic, the leading harmonic is the dominant term
in the Fourier expansions of the zero-point kinetic energy and Zeeman energy.
As a result, the leading harmonic expansion accurately captures the trend of
the bandwidth and quantum geometry, though it may fail to quantitatively
reproduce more detailed information about the bands such as the Berry curvature
distribution.
