A time-based Chern number in periodically-driven systems in the adiabatic limit

TL;DR

This paper establishes a relationship between the synthetic 2D Chern number in periodically driven systems and the Chern number from a parametric variable, revealing how to engineer topological invariants via parametric evolution.

Contribution

It demonstrates that in adiabatic periodically driven systems, the synthetic 2D Chern number is proportional to the Chern number of a parametric variable, linking synthetic and physical topological invariants.

Findings

Synthetic 2D Chern number is a multiple of the parametric Chern number.

The relationship is validated through Thouless pumping in two models.

Findings can be extended to higher dimensions and other configurations.

Abstract

To define the topology of driven systems, recent works have proposed synthetic dimensions as a way to uncover the underlying parameter space of topological invariants. Using time as a synthetic dimension, together with a momentum dimension, gives access to a synthetic 2D Chern number. It is, however, still unclear how the synthetic 2D Chern number is related to the Chern number that is defined from a parametric variable that evolves with time. Here we show that in periodically driven systems in the adiabatic limit, the synthetic 2D Chern number is a multiple of the Chern number defined from the parametric variable. The synthetic 2D Chern number can thus be engineered via how the parametric variable evolves in its own space. We justify our claims by investigating Thouless pumping in two 1D tight-binding models, a three-site chain model and a two-1D-sliding-chains model. The present findings could be extended to higher dimensions and other periodically driven configurations.