Fractionally Quantized Electric Polarization and Discrete Shift of Crystalline Fractional Chern Insulators

Fractional Chern insulators (FCI) with crystalline symmetry possess topological invariants that fundamentally have no analog in continuum fractional quantum Hall (FQH) states. Here we demonstrate through numerical calculations on model wave functions that FCIs possess a fractionally quantized electric polarization, ${\vec{\mathcal{P}}}_{\text{o}}$, where $\text{o}$ is a high symmetry point. ${\vec{\mathcal{P}}}_{\text{o}}$ takes fractional values as compared to the allowed values for integer Chern insulators because of the possibility that anyons carry fractional quantum numbers under lattice translation symmetries. ${\vec{\mathcal{P}}}_{\text{o}}$, together with the discrete shift ${\mathcal{S}}_{\text{o}}$, determine fractionally quantized universal contributions to electric charge in regions containing lattice disclinations, dislocations, boundaries, and/or corners, and which are fractions of the minimal anyon charge. We demonstrate how these invariants can be extracted using Monte Carlo computations on model wave functions with lattice defects for 1/2-Laughlin and 1/3-Laughlin FCIs on the square and honeycomb lattice, respectively, obtained using the parton construction. These results comprise a class of fractionally quantized response properties of topologically ordered states that go beyond the known ones discovered over thirty years ago.