# Measuring Chern numbers in Hofstadter strips

## TL;DR

This paper demonstrates a method to measure Chern numbers in Hofstadter strips with open boundaries and limited size by observing atomic wavepacket displacement under a weak force, enabling topological characterization in small systems.

## Contribution

It introduces a scheme to measure Chern numbers in small Hofstadter strips with open boundary conditions, even with as few as three sites, using atomic transport under a weak force.

## Key findings

- Transverse displacement approaches the quantized Chern number after one Bloch oscillation.
- The method maps out Chern numbers for both ground and excited bands.
- Robustness of the measurement is confirmed against disorder and harmonic trapping.

## Abstract

Topologically non-trivial Hamiltonians with periodic boundary conditions are characterized by strictly quantized invariants. Open questions and fundamental challenges concern their existence, and the possibility of measuring them in systems with open boundary conditions and limited spatial extension. Here, we consider transport in Hofstadter strips, that is, two-dimensional lattices pierced by a uniform magnetic flux which extend over few sites in one of the spatial dimensions. As we show, an atomic wavepacket exhibits a transverse displacement under the action of a weak constant force. After one Bloch oscillation, this displacement approaches the quantized Chern number of the periodic system in the limit of vanishing tunneling along the transverse direction. We further demonstrate that this scheme is able to map out the Chern number of ground and excited bands, and we investigate the robustness of the method in presence of both disorder and harmonic trapping. Our results prove that topological invariants can be measured in Hofstadter strips with open boundary conditions and as few as three sites along one direction.