Crystalline invariants of fractional Chern insulators

TL;DR

This paper develops a theoretical framework and numerical methods to identify crystalline invariants in fractional Chern insulators, linking topological order, symmetry fractionalization, and measurable ground state properties.

Contribution

It introduces a new approach using partial rotation expectation values to extract crystalline invariants applicable to both Abelian and non-Abelian topological orders.

Findings

Theoretical invariants match numerical Monte Carlo results.

Hall conductivity, filling fraction, and partial rotation invariants fully characterize crystalline invariants.

Framework applies to continuum fractional quantum Hall states with rotational symmetry.

Abstract

In the presence of crystalline symmetry, topologically ordered states can acquire a host of symmetry-protected invariants. These determine the patterns of crystalline symmetry fractionalization of the anyons in addition to fractionally quantized responses to lattice defects. Here we show how ground state expectation values of partial rotations centered at high symmetry points can be used to extract crystalline invariants. Using methods from conformal field theory and G-crossed braided tensor categories, we develop a theory of invariants obtained from partial rotations, which apply to both Abelian and non-Abelian topological orders. We then perform numerical Monte Carlo calculations for projected parton wave functions of fractional Chern insulators, demonstrating remarkable agreement between theory and numerics. For the topological orders we consider, we show that the Hall conductivity, filling fraction, and partial rotation invariants fully characterize the crystalline invariants of the system. Our results also yield invariants of continuum fractional quantum Hall states protected by spatial rotational symmetry.