Topological invariants in two-dimensional quasicrystals

TL;DR

This paper introduces a topological framework for characterizing energy gaps in two-dimensional quasiperiodic systems, linking them to second Chern numbers and the four-dimensional quantum Hall effect, regardless of periodicity commensurability.

Contribution

It provides a novel topological classification of energy gaps in 2D quasiperiodic systems using second Chern numbers derived from adiabatic charge pumping.

Findings

Energy gaps are characterized by integer sets quantizing momentum space areas.

These integers are expressed as second Chern numbers related to 4D quantum Hall effect.

Topological invariants are independent of periodicity commensurability and deformation.

Abstract

We study the topological characterization of the energy gaps in general two-dimensional quasiperiodic systems consisting of multiple periodicities, represented by twisted two-dimensional materials. We show that every single gap is uniquely characterized by a set of integers, which quantize the area of the momentum space in units of multiple Brillouin zones defined in the redundant periodicities. These integers can be expressed as the second Chern numbers, by considering an adiabatic charge pumping under a relative slide of different periodicities, and using a formal relationship to the four-dimensional quantum Hall effect. The integers are independent of commensurability of the multiple periods, and invariant under arbitrary continuous deformations such as a relative rotation of twisted periodicities.