Crystalline topological invariants in quantum many-body systems

TL;DR

This paper reviews recent advances in understanding crystalline symmetry-protected topological invariants in quantum many-body systems, especially in strongly interacting and lattice models like Chern insulators.

Contribution

It synthesizes recent methods for characterizing, classifying, and detecting crystalline topological invariants in two-dimensional quantum systems.

Findings

Crystalline symmetries protect topological invariants in quantum phases.

Recent non-perturbative methods reveal invariants in strongly interacting models.

Lattice translation and rotation symmetries are key to classifying these invariants.

Abstract

Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent developments have demonstrated that even classic models, like the Harper-Hofstadter model of free fermions on a lattice in a magnetic field, yield a host of crystalline symmetry protected topological invariants. Here we review some of these developments, focusing mainly on how to characterize, classify, and detect invariants arising from lattice translation and rotation symmetries along with charge conservation in two-dimensional systems, including integer and fractional Chern insulators.