Quasihole Dispersion from Quantum Geometry in Fractional Quantum Anomalous Hall States

The discovery of fractional quantum anomalous Hall (FQAH) states in twisted TMDs and rhombohedral graphene opened the door to the exciting possibility of realizing phases of dispersing anyons. However, computing the magnitude of this dispersion and how it depends on microscopic parameters remains a challenge. Here, we show that by considering FQAH states in ideal or Aharonov-Casher (AC) bands, we can develop an analytically controlled theory of anyon dispersion by projecting the interaction on the space of Laughlin quasiholes. We construct quasihole momentum eigenstates and develop a momentum space quantum Monte Carlo approach which enables us to study the dispersion for large system sizes. We identify the origin of quasihole dispersion as arising from a combination of the non-uniform quantum geometry of the bands that generate a periodic potential for the quasiholes and a "quasihole quantum geometry" which turns this periodic potential into a dispersion. We develop a microscopic Lagrangian framework in terms of a quasihole guiding center coordinate that describes the dynamics of the quasihole and reproduce the momentum space results. Our framework paves the way to the development of a fully microscopic theory of quasiholes in FQAH which is minimal in the sense that it only accounts for the quasihole degrees of freedom.