Locality of topological dynamics in Chern insulators.

This paper provides a comparative analysis of the properties of two local topological markers under real time dynamics. These markers are the local Chern marker (LCM) C(r,t) (calculated through a single-particle projection of occupied states to a single site), and the Streda marker Cs(r,t) (which corresponds to the excess charge induced by an increase in magnetic field). In equilibrium, the bulk average of C and Cs is the global topological invariant known as the Chern number. Yet, as pointed out by the authors, C and Cs do not generally coincide, even in equilibrium.

Focusing first on the LCM's dynamics, the authors identify two contributions to the currents driving the time-evolution of C(r,t): a local term, which dominates in large translationally invariant patches, and a non-local contribution (originating from the non-locality of projection operators). Regarding the local Streda marker, the authors take experimental consistency into account to define a local, instantaneous quantity which may be measured in experiments. This definition leads to a local Streda marker governed by a continuity equation. The authors then turn to concrete examples, and numerically study the quench dynamics of C and Cs in various topological models with various scenarios, such as a global quench (from topological to trivial) in a translationally invariant model or a slow shift of a topological domain over time. This study confirms and illustrates their previous analytical observations.

Overall, the main result of the paper lies in the identification of the regimes where the markers take similar values, and in the analysis of their difference. I find this to be a very valuable contribution. In particular, this study provides a useful benchmark for situations where the direct application of the LCM formula is not possible, such as experiments or even the theoretical study of strongly interacting models. The experimental relevance is highlighted by the fact that the Streda marker may be measured in AMO experiments, or in some solide state ones using SQUID.

I therefore recommend publication, provided that the authors can answer the following questions and fix a few issues:

- A lot of information on the figure is too small to read (enclosed captions in fig.2 and 3 in particular).

- What do the colored dots in fig. 3e indicate?

- In fig.4, the authors indicate that the discrepancies between C and Cs correspond to times where the reflected propagation front comes back to the original site. This interpretation seems to work well at small times, where reflections can be identified in fig.4c. How can we understand the correpondence between C and Cs at larger times (t ~ 60 - 80), and the role played by the number of bulk sites over which the marker is averaged?

- Finally, to increase the experimental relevance of this study, it would be interesting to include a discussion of relevant time scales: e.g. time associated with the variation of microscopic parameters, or with the variation of the magnetic flux in the Streda formula. How do these time scales compare to experimental standards?

Markov et al pose the pertinent question of what is the best definition for a time-dependent local marker. In their work, they compare different alternatives based on the definition of the local Chern marker and a local Streda marker. The two differ already in equilibrium and thus it is pertinent to ask when and how these differences matter out of equilibrium. They study several examples and find that translational invariance is key for their equivalence.

While I consider the work interesting I have some comments for the authors to consider before I can recommend publication.

First, they claim after eq 13 that the local Streda marker is a better estimation for the time-dependent local Chern marker. I am not sure it is clear from the text around, or the rest of the paper what criterion for “better” the authors mean. Can the author’s clarify what is better about one over the other? The paper shows they are in general different, but why one is “better” than the other is unclear to me.

Perhaps related to the above question: Is C_S(t) a more experimentally realistic observable than C(t)?

One puzzling (or subtle in the author’s words) statement in the paper concerns the gauge dependence in page 12. Are the contributions Jc and M independently measurable and gauge independent? If not, wouldn’t their distinction be arbitrary and this can cause difficulties.

At any rate I find confusing to say that there are two situations that are not “gauge equivalent". It is seems to me (as seem to be the case in Ref. [37]) that the difference of the “gauge choice” amounts to the taking into account the fact that A, the gauge potential, depends on t, and this leads to an electric field. I am unsure though if calling the two situations "gauge non-equivalent” is fair, as they constitute two different physical situations.

My last comment concerns the role of translational invariance. While it is evident from the authors results that it plays a role, what is the physical intuition? is it that the local Chern marker in this case ceases to be quasi-local?

Minor Comments:

— Typo above eq. 11 “applying magnetic field to an initial state and when allow…” when -> then?
— bulk appears twice in the first paragraph of section 4. “…translation invariance of the bulk bulk…”
— Figure 2 b at t=13 and t=19, C and C_S seem to have a certain degree of anti-correlation. Can the authors comment if this is a generic or accidental feature?
— I find that the figures can be greatly improved. In Fig. 3 c there are no labels for the axis or the color map of the insets.
— Some figures are not referred to at all in the main text, which makes their appearance questionable.
— w.r.t. can be spelled out.
— there is a missing punctuation mark in appendix D, end of second sentence.