Anyon Wigner crystals and phase competition in fractional quantum hall and Chern insulators

Fractional quantum Hall (FQH) and fractional Chern insulator (FCI) phases host exotic topological order with anyonic excitations. We investigate crystalline order in many-anyon systems by quantizing semiclassical anyon dynamics. For states with many anyonic holes in the Laughlin (filling fraction = \nu = 1/m) vaccum, we use the plasma analogy to construct harmonic approximation Lagrangians of anyon crystal states formed in both FQH and FCI states. Monte Carlo sampling is used to compute energies and Berry curvature as a function of quasi-hole positions. We use the harmonic theory of anyon dynamics to calculate phonon spectrs and the entropy of the resulting Wigner crystal. We identify regimes in which anyonic crystal states compete with electron or hole crystal states, providing new insight into correlated Chern band phenomena.