# Probing chiral edge dynamics and bulk topology of a synthetic Hall system

## Abstract

Quantum Hall systems are characterized by quantization of the Hall conductance—a bulk property rooted in the topological structure of the underlying quantum states1. In condensed matter devices, material imperfections hinder a direct connection to simple topological models2,3. Artificial systems, such as photonic platforms4 or cold atomic gases5, open novel possibilities by enabling specific probes of topology6,7,8,9,10,11,12,13 or flexible manipulation, for example using synthetic dimensions14,15,16,17,18,19,20,21. However, the relevance of topological properties requires the notion of a bulk, which was missing in previous works using synthetic dimensions of limited sizes. Here, we realize a quantum Hall system using ultracold dysprosium atoms in a two-dimensional geometry formed by one spatial dimension and one synthetic dimension encoded in the atomic spin J = 8. We demonstrate that the large number of magnetic sublevels leads to distinct bulk and edge behaviours. Furthermore, we measure the Hall drift and reconstruct the local Chern marker, an observable that has remained, so far, experimentally inaccessible22. In the centre of the synthetic dimension—a bulk of 11 states out of 17—the Chern marker reaches 98(5)% of the quantized value expected for a topological system. Our findings pave the way towards the realization of topological many-body phases.

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## Data availability

Source data, as well as other datasets generated and analysed during the current study, are available from the corresponding author upon request. Source data are provided with this paper.

## Code availability

The source code for the numerical simulations of the Abrikosov vortex lattices and the Laughlin states are available from the corresponding author upon request. Source data are provided with this paper.

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## Acknowledgements

We thank J. Beugnon, N. Cooper, P. Delpace, N. Goldman, L. Mazza and H. Price for stimulating discussions. We acknowledge funding from the EU under ERC projects ‘UQUAM’ and ‘TOPODY’, and PSL research university under the project ‘MAFAG’.

## Author information

Authors

### Contributions

All authors contributed to the set-up of the experiment, data acquisition, data analysis and the writing of the manuscript.

### Corresponding author

Correspondence to Sylvain Nascimbene.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Conservation of canonical momentum.

Difference between the measured canonical momentum $${p}_{{\rm{rec}}}\langle {\hat{J}}_{z}\rangle +M\langle \hat{v}\rangle$$ and the targeted value p defined by the state preparation protocol. All error bars are the 1-σ standard deviation of typically 5 measurement repetitions.

### Extended Data Fig. 2 Cyclotron orbits measurements.

a, b, c, Magnetization, velocity and position response as a function of time after application of a velocity kick vkickvrec. d, Bulk excitations corresponding to different velocity kicks, v = 0.23, 0.48, 1.02, 2.06, and 3.15 vrec, from left to right. The orbits are off-centred in real space for visual clarity. e, Skipping orbit for the momentum state p = − J prec following a sudden jump of the coupling strength Ω. f, Velocity drift of the orbits as a function of the amplitude kick. The solid line corresponds to the expected drift due to the non-harmonic spectrum of the energy bands. All error bars are the 1-σ standard deviation of typically 5 measurement repetitions.

### Extended Data Fig. 3 Hall system in real dimensions.

a, Variation of the Hall mobility for classical skipping orbits, depending on the angle of rebound on a hard wall. The case of closed cyclotron orbits corresponds to θ = π. b, Dispersion relation of a quantum Hall system in a semi-infinite geometry y > 0. The blue line indicates the energy branch used for the Chern marker calculation, defined by E0(p) < ωc. c, Hall mobility μ as a function of momentum p. d, Local density of state in the (v, y) plane. e, Local Chern marker C(y) for the energy branch defined in b.

### Extended Data Fig. 4 Hall mobility and local Chern markers.

a, Predicted dispersion relation for Ω = Erec. The branch pictured in blue, chosen as E(p) < E* with E* at half the gap, is used for the computation of the local Chern marker. b, Measured mobility in x resulting from the application of a force along m, as presented in the main text. The points in blue, corresponding to p < p* (white area), are the ones considered for the Chern marker presented in the main text (see Fig. 4). c, Measured mobility in m resulting from the application of a force along x. As for b, the points in red are associated to momentum states lying below E*. d, Chern marker obtained from the measured mobility, using the whole energy branch (− < p < , gray squares, using data in b), or using the branch defined in a (− p* < p < p*). For the latter, the blue dots correspond to the data in b, and are identical to Fig. 4. The red diamonds correspond to the data in c. Solid lines are theoretical values. The error bars are the 1-σ statistical uncertainty calculated from a bootstrap sampling analysis over typically 100 pictures (b,c) and 1000 pictures (d).

### Extended Data Fig. 5 Effect of disorder.

a, Example of Chern marker distribution in the presence of disorder of strength Δ = Erec. b, Chern marker $$\bar{C}(m=0)$$ averaged over the region x < λ/4 as a function of the disorder strength Δ. Each point is the average of 100 disorder realizations, the error bar showing the standard deviation of the mean.

### Extended Data Fig. 6 Abrikosov vortex lattices.

a, Ground state density profile and b, associated phase, for Ω = 3Erec and for μchem ≈ 4Erec. The local minima of the density exhibit a phase winding around them, and thus correspond to quantum vortices. c, Number of vortex lines as a function of the Raman coupling Ω and the chemical potential μchem. The dots identify the configurations for which a simulation was realized. The color encodes the number of vortex lines that characterizes the low-energy vortex lattice configuration. The phase separation lines are guides to the eye. The dashed line identifies the gap to the first excited band above which the atoms significantly occupy higher Landau levels. d, Momentum p0 associated to the spontaneous breaking of the translational invariance resulting from the appearance of a vortex lattice, as a function of Ω. The points were taken at a chemical potential corresponding to half the gap.

## Supplementary information

### Supplementary Information

Supplementary Discussion and Figs. 1–5.

## Source data

### Source Data Fig. 2

Source data for panels a and b.

Source data.

### Source Data Fig. 4

Source data for panels b and c.

### Source Data Fig. 5

Source data for panel c.

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Chalopin, T., Satoor, T., Evrard, A. et al. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-0942-5

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