Fractional Chern insulators (FCIs) with crystalline symmetry possess topological invariants that fundamentally have no analog in continuum fractional quantum Hall 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 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 ${\mathcal{S}}_{\text{o}}$, determines fractionally quantized universal contributions to electric charge in regions containing lattice disclinations, dislocations, boundaries, and/or corners, 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 lattices, 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 30 years ago.

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• Chern insulators
• Edge states
• Electric polarization
• Fractional quantum Hall effect
• Symmetry protected topological states
• Topological defects
• Topological field theories