Anyon Dispersion from Nonuniform Magnetic Field on the Sphere

The discovery of fractional quantum anomalous Hall states in moiré systems has raised the possibility of realizing phases of itinerant anyons, whose mobility requires the absence of continuous magnetic translation symmetry (CMTS). Motivated by this, we consider anyons on the sphere in the presence of a nonuniform magnetic field that breaks the SU(2) rotation symmetry, the analog of CMTS on the sphere, down to a U(1). This allows us to study the energy dispersion of the anyons as a function of the angular momentum L_{z}, while maintaining the perfect flatness of the single-particle dispersion. We parametrize the nonuniform field by a real parameter R that concentrates the field at the north (south) pole for R>1 (R<1). For this choice of field, any p-body correlation function evaluated in the space of Laughlin quasiholes maps exactly onto the corresponding p-body correlation function in uniform field. In the thermodynamic limit, this exact mapping lets us analytically compute the interaction-generated, spatially varying potential felt by the anyons. Remarkably, this static potential alone is sufficient to generate anyon dispersion: because each anyon experiences an emergent background magnetic field, the two components of its position do not commute, so the potential is converted into a finite dispersion, which we compute exactly up to an overall scaling constant. The resulting anyon dispersion describes the azimuthal motion around the sphere at constant height, the analog of spin precession. Our Letter thus provides an exactly solvable realization of a general mechanism for interaction-induced anyon dispersion-a symmetry-breaking potential converted into kinetic energy by the anyon's many-body Berry phase-with no single-particle dispersion or hopping of any kind.