Anyon dispersion from non-uniform magnetic field on the sphere

Abstract

The discovery of fractional quantum anomalous Hall states in moir\'e systems has raised the interesting possibility of realizing phases of itenerant anyons. Anyon dispersion is only possible in the absence of continuous magnetic translation symmetry (CMTS). Motivated by this, we consider anyons on the sphere in the presence of a non-uniform magnetic field which breaks the ${\rm SU}(2)$ rotation symmetry, the analog of CMTS on the sphere, down to a ${\rm U}(1)$. This allows us to study the energy dispersion of the anyons as a function of $L_z$ angular-momentum, while maintaining the perfect flatness of single-particle dispersion. We parametrize the non-uniform field by a real parameter $R$ which concentrates the field at the north (south) pole for $R>1$ ($R < 1$), and show that, for our choice of field, any $p$-body correlation function evaluated in the space of Laughlin quasiholes can be mapped \emph{exactly} to a corresponding $p$-body correlation function in uniform field. In the thermodynamic limit, this enables us to analytically compute the interaction-generated spatially varying potential felt by the anyons. Remarkably, such spatially varying potential is sufficient to generate dispersion for the anyons, which we compute exactly, up to an overall scaling constant. The anyon dispersion in our model describes azimuthal motion around the sphere at a constant height, similar to spin precession. Our work therefore serves as a concrete demonstration that interaction alone can generate nonzero anyon dispersion in the presence of inhomogeneous magnetic field.