Adiabatic Approximation and Aharonov-Casher Bands in Twisted Homobilayer TMDs

TL;DR

This paper investigates the validity of the adiabatic approximation in twisted homobilayer TMDs and demonstrates that Aharonov-Casher bands, which are crucial for ideal quantum geometry, emerge under certain parameter regimes, with the approximation being generally accurate.

Contribution

It critically examines the adiabatic approximation in twisted TMDs and clarifies the conditions for the emergence of Aharonov-Casher bands, highlighting the importance of the leading Fourier harmonic.

Findings

Adiabatic approximation is valid across many experimental parameters.

Leading Fourier harmonic dominates the zero-point kinetic and Zeeman energies.

The approximation captures trends in bandwidth and quantum geometry but not detailed Berry curvature distributions.

Abstract

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.