The discovery of fractional quantum anomalous Hall (FQAH) states in two-dimensional heterostructures has opened the door to realizing phases of dispersing anyons. Here, we develop an analytically controlled theory of anyon dispersion in FQAH states realized in ideal or Aharonov-Casher (AC) bands by projecting interactions onto the space of Laughlin quasiholes. Constructing quasihole momentum eigenstates allows efficient evaluation of the single-quasihole dispersion using Monte Carlo. We find that the quasihole bandwidth grows with increasing quantum-geometry inhomogeneity of the AC band and with increasing interaction screening length. For realistic parameters relevant to the bands of twisted $\mathrm{MoTe}{}_2$, the quasihole bandwidth is of order 1 meV and increases with increasing displacement field, suggesting that itinerant-anyon physics may play an important role in sufficiently clean samples. Furthermore, we develop a microscopic Lagrangian framework in terms of a quasihole guiding-center coordinate, which reproduces the momentum-space formula for the dispersion. This approach reveals that quasihole dispersion originates from the combined effects of an interaction-generated periodic potential, arising from nonuniform quantum geometry of the single-particle bands, and the quasihole many-body Berry phase arising from the background magnetic field. The latter endows the guiding-center coordinate with a noncommutative structure, converting the periodic potential into a finite dispersion. Finally, we outline how this framework generalizes to multiple quasiholes, enabling a microscopic theory of charged excitations in FQAH systems that retains only the anyon degrees of freedom.

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• Anyons
• Electronic structure
• Flat bands
• Landau levels
• Quantum geometry
• Quantum Monte Carlo