Measuring Chern numbers in Hofstadter strips

1. The authors propose a novel way to extract the topological Chern number from the transport of a wave-packet in a narrow Hofstadter strip. This is a very timely and interesting proposal, which opens up new perspectives for measuring topological invariants in small systems.

2. This proposal seems realistic from an experimental point-of-view, and it nicely exploits the advantages of using synthetic dimensions in ultracold atoms. The authors also support their proposal by discussing the robustness of their method under harmonic trapping and disorder.

3. This paper is clearly explained and should be accessible to many people in the field. It also presents an understanding of Chern number physics which is quite intuitive while also being complementary to the current literature.

4. The strength of this method is further demonstrated by its application to measure higher Chern numbers and to even beat the FHS algorithm, which is a much used tool in the field.

1. There are a few issues that need to be clarified within the manuscript. These are listed as Requested Changes below.

In this paper, the authors propose to measure topological Chern numbers from the dynamics of a wave-packet in a narrow Hofstadter strip: a two-dimensional lattice, pierced by a uniform magnetic flux, which is restricted to only a few sites along one of the spatial directions. This proposal is directly relevant to cutting-edge ultracold gas experiments on so-called synthetic dimensions built out of internal atomic states. Due to practical limitations on the number of internal atomic states coupled, these experiments realised Hofstadter strips and explored quantum Hall edge physics; this paper now proposes how to go further to also probe the topological bulk properties associated with a fully-extended 2D system. This is therefore a timely and interesting proposal, which can be helpful to experimentalists in the field.

Overall, this paper is well-written, accessible and well-explained. The authors have also cited widely, so that I think this paper can also serve as a valuable starting point for an interested reader to explore the rest of the field. I recommend publication of this paper after a few small issues, listed in Requested Changes, have been resolved.

1. When introducing synthetic lattices on pg. 2, the authors specify "systems where particles have D spatial degrees of freedom and an extra synthetic dimension" but then later in the paragraph talk about "4D models", which implies another definition of "D"?

2. Refs. [45-47] are cited as experiments on pg. 2, but these are theoretical proposals. I would recommend adding these instead to "...optical resonators [49]".

2. In Figure 1a) the blue arrow is very hard to see. I would also suggest adding axes labels for Fig 1.b) to be clear. It would also be helpful to the reader to explain the coloring of different sites in Fig. 1b).

3. As a question to the authors: how can we see that all quasi-momentum states are populated equally over this definition of a period of the force (Eq. 10)? From Section II B and as implied by Fig. 3, the chosen Brillouin zone is $(2 \pi)/d \times (2 \pi/q d)$ along $k_x$ and $k_y$ respectively, which implies that you need to wait $T=2 \pi \hbar / d |F_x|$ to change $k_x$ by $2\pi$ and to sweep through the whole MBZ?

4. Could the authors please comment e.g. in Sec IV A on how they choose the force relative to the hybridisation gap? What size is the hybridisation gap here?

5. In Sec IV A, the authors say "the Hamiltonian with open boundary conditions and zero force, Eq. (3)". Do they mean Eq. 1?

6. Could the authors comment on how, when they project the evolved state on the eigenvectors of the Hamiltonian with open boundary conditions, they make the distinction between band and gap states in Fig 4 b)?

7. In Sec. IV C1, the authors refer to "point A in Fig. 2a" but I do not see point A in this figure?

8. What is the strength of the force used in Section IV D? In the inset of Fig. 7, the hybridisation gap looks very close to the energy band gap so it is possible to chose a good value for the force between these limits?

9. In Fig. 7, I am curious about the identification of "well localised edge states crossing at $k_x = 3 \pi / 5d$": according to the color scheme, I see that the state with a positive gradient is well-localised at $<y>=-2d$, but the state with a negative gradient does not seem to be localised as I would have expected around $<y>=2d$. Why can this second state be called an edge state?

10. Question to the authors: in Fig. 7 & 8, it is shown that the Chern number of the lowest band can be extracted even when the number of eigenstates is smaller than the number of bands in the corresponding extended model. If instead the experiment was started from the minimum at $k_x=-4 \pi/5d$ then would this still work? Could the authors comment on how general these results are expected to be?

1. The authors propose and analyse a novel method enabling to determine topological properties of the bulk by measuring the dynamics of an wave-packet in a Hofstadter stripe.
2. This is can be useful, e.g., to cold atom experiments where the magnetic flux can be produced using synthetic dimensions. In that case the stripe naturally appears, because the synthetic dimension extends over a limited number of sites.

1. The manuscript deals only with the single particle physics. I suppose the many-body aspects of this method will be studied later on, as in the Concluding Section VI it is written:
"Another interesting direction is to analyze the effect of strong interactions on our protocol [60, 69], specially in view of the recent theoretical interest on interacting phases and Laughlin-like states in real and synthetic ultracold atom ladders [97–100]."

The authors consider the transport of particles in two-dimensional lattices affected by a uniform magnetic. It is shown that the topological properties of the bulk can be determined from the dynamics of an wave-packet in a Hofstadter stripe taken from the bulk. This is relevant for example to cold atom experiments where the magnetic flux can be produced using synthetic dimensions. In that case the stripe naturally appears, because the synthetic dimension extends over a limited number of sites.

The authors have explored the dynamics of a wave-packet which is initially localized in the transverse direction of the stripe. In the context of the semi-synthetic lattices such a wave-packet can be produced by initially placing atoms in a single internal state: this corresponds to a perfect localization in the synthetic dimension. Subsequently a weak constant force is applied in the real dimension leading to the Bloch oscillations. It is shown that after one Bloch cycle, the wave-packet acquires a displacement in the transverse (synthetic) direction determined by the quantized Chern number of the infinite system. The scheme enables to determine the Chern numbers of both the ground and excited bands. The robustness of the proposed method with respect to the disorder and harmonic trapping has also been investigated.

The paper is well written. I think it is an interesting work, which is relevant to the current experiments with cold atom, photonic and condensed matter systems. For example, applying the proposed method the Chern number of the bulk can be measured using the stripes containing as few as three sites in transverse direction. This could be very helpful in studying the semi-synthetic ribbons. I recommend the manuscript to be published.

A minor remark. The method considered by the authors enables one to measure the first Chern number. The authors could discuss the extension of the method to measure of the second Chern number. In the last sentence of the Concluding Section VI there is some hint in this direction, but the authors could be more explicit.

See the end of the Report