A dark state of Chern bands: Designing flat bands with higher Chern number

Dear Editors,

We hereby resubmit our manuscript entitled "A dark state of Chern bands: Designing flat bands with higher Chern number".

We appreciated the very constructive and thoughtful reports of the three Referees, which we hereby acknowledge. A reasonable list of questions and suggestions for improvements were formulated in these reports, which invited us to perform a substantial revision of our work.

Most importantly, our revised manuscript now contains: (a) a mathematical proof of the sum rule [Eq. 9], which constitutes the central result of our work; (b) new numerical simulations of the loading and detection schemes, which are now compatible with realistic experimental time scales; (c) a clearer and more pedagogical description of the Lambda system.

A point-by-point answer to the Referees' comments are provided below, as well as a list of changes.

We believe that the revised manuscript fully addresses all the concerns and suggestions of the Referees, making it suitable for publication.

N. Goldman, on behalf of the authors.

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Reply to Referee 1:

We thank Referee 1 for their careful reading of our work. We are very pleased that the Referee finds our work interesting and suitable for publication. We are also very grateful for their insightful remarks regarding experimental aspects.

The requested changes concerned the timescales of the proposed protocol in view of experimental implementation. We have thoughtfully addressed this issue in the revised manuscript, which presents new simulations of the loading protocol and Chern-number measurement. In particular, Table 1 now explicitly presents the error (transfer to the undesired states) for various system sizes, ramp durations and other system parameters. The main results shown in Fig. 3 now also correspond to more realistic time scales (<1s), compatible with experimental coherence times. We note that a new ramp profile has been designed in view of optimizing the protocol (allowing for ramp durations of the order of 10^3 hbar/J); we also improved the duration of the Chern-number measurement (now also of the order of 10^3 hbar/J) by increasing the strength of the Rabi couplings. Under these conditions, the total Hall drift corresponds to about 30 lattice sites, which is satisfactory in view of detection.

We believe that our revised manuscript takes all the suggestions of the Referee into account, and that it is now in a suitable form for publication.

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Reply to Referee 2:

We thank Referee 2 for their careful reading of our manuscript. We are pleased that the Referee finds our work interesting and suitable for publication.

Following a suggestion of the Referee, we now moved some informations from the Appendix to the main text. In particular, this should clarify some important notions and definitions inherent to our \Lambda scheme (see Section 2 and 3).

We now address the remarks of the Referee below:

1) "While it is understandable that Eq. (6) produces one dark band and two bright bands, its connection to the previous Lambda system is not clear. Here all the three bands are overlapping without the coupling As, thus it doesn't look like a Lambda-type."

As for a standard \Lambda system, where three states at different energies are coupled resonantly by time-dependent fields, we move to a rotating frame and apply the rotating wave approximation; after this operation, the three states are at the same energy (up to some detuning) and they are coupled by time-independent fields. This is precisely the situation described in Eq. (6). We now clarify this in the revised manuscript [see Eqs. 1-2 and Appendix B].

2) "What is the value of As in Fig. 2(c), and its relation to Ωs(k) (in the caption of Fig. 2)?"

We thank the Referee for this relevant question. The Ωs(k) are the matrix elements connecting the states 1,2 and 3, e.g \Omega_1^*= i <1(k) | H | 3(k)>; we now explicitly define this quantity in the revised manuscript [see Eqs. 7-8, as well as the new introductory Section 2]. Considering the multi-species configuration Hamiltonian in Eq. (6) [now Eq. 13!], which is illustrated in Fig. 2, the Ωs are set by the coupling terms (second sum in Eq. 13) and they are thus directly proportional to As in Eq. 13. The actual values of As are not very instructive (as they strongly depend on the system's detail), hence we opted to provide the maximal value of the effective couplings Ωs instead (which directly enter the common lambda-system description); see caption of Fig. 2.

3) "Finally, why is the trivial band so flat? If naively H_HH with phi=0 is taken, the bandwidth is \sim J. Is it due to the fact that only every q sites are used along the x-axis, which implies Jx \ll Jy?"

We also thank the Referee for this question. The trivial band is taken from a very deep state-dependent lattice, whose bandwidth is assumed to be negligible as compared to all other energy scales. We emphasize that this bandwidth is not related to the parameter J entering the Hofstadter model in Eq. (5), as it stems from a different (state-dependent) lattice. We now fully clarify this point in the revised version (above Eq. 13).

We believe that our revised manuscript takes all the suggestions of the Referee into account, and that it is now in a suitable form for publication.

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Reply to Referee 3:

We thank Referee 3 for their careful reading of our work. We are very pleased that the Referee finds our results new and interesting.

The Referee's main criticism concerned the fact that there was "very little intuition behind the main result " (i.e. the sum rule CD=C1+C2-C3). This relevant criticism motivated us to develop an analytical proof of this sum rule, which we now proudly present in our revised manuscript. This mathematical proof, which is based on the introduction of a well-defined homotopy (see Eqs. 9 to 12), represents a major improvement and extension of our manuscript.

In particular, the addition of this proof simultaneously and rigorously addresses several questions and criticisms of the Referee. For instance, this proof indicates that the sum rule still holds even if the light-coupling carries angular momentum; it explains why the third state contributes to the Chern number, and why the phase between states 1 and 2 does not contribute.

Regarding the question "why do we need three levels":

First of all, we emphasize that our construction entirely relies on the notion of the dark state, which allows to generate a flat dark band with predictable Chern number (as dictated by the simple sum rule involving the bare bands’s Chern numbers). Our dark-state construction requires at least three bands, by definition of the dark state.

That being said, one can study the fate of two bare bands (1,2), resonantly coupled by some driving fields; this configuration would lead to two dressed bands (\pm), and the Chern numbers would satisfy (C+) + (C-) = C1 + C2, whenever the dressed bands are separated by a gap (i.e. whenever the Chern numbers are well defined). Now, a simple calculation shows that considering two bare bands with different Chern numbers (C1 \ne C2) naturally leads to a gap closing point within the Brillouin zone (for a drive that is sufficiently strong to potentially change the initial Chern numbers); hence this situation cannot be considered to generate dressed bands with tunable Chern numbers. One is thus left with the scenario where C1=C2. We note that C1=C2=0 is actually the starting point for the construction of Haldane-type models through Floquet engineering (see, for instance, the experiments in the group of Sengstock and Weitenberg in Hamburg). However, in that case, it is not possible to: (a) simply design a flat dressed band, and (b) to offer an intuitive prediction for the dressed band's Chern numbers for a general driving scheme. This motivates the use of the dark-state approach (using more than two bands), as explained above. We now further motivate our approach in the abstract and introduction.

Regarding the application to the five-state M-scheme: a similar proof can indeed be obtained for the sum-rule formula, which would indeed take the form proposed by the Referee: CD=C_1+C_3+C_5-C_2-C_4. This result is based on the observation that the four bright states can be expressed in a form similar to Eq. (4); applying a similar homotopy argument as for the 3-state system, one obtains CD=(C1+...+C5) - 2C2 - 2C4 = C1+C3+C5-C2-C4.

Finally, the Referee suggested that our work "seems very similar to Panas2020". This remark motivated us to fully clarify -- in the manuscript -- the main and crucial differences between the results of Panas2020 [Ref. 63] and our work (see the last paragraph of Section 3). We hereby reproduce this added paragraph:

"We note that similar sum rules were numerically investigated in the multilayer configuration of Ref. [63]. The latter work considered the stacking of trivial (boron-nitride-type) and non-trivial (Haldane-type) 2D lattices, and it explored the total Chern number of the system at half-filling as a function of the inter-layer coupling strength. In that context, the calculated Chern number reflects the topology of a three-fold degenerate band, which is associated with the three underlying layers; using an open-system approach, this total Chern number was decomposed as a sum of sub-system indices [64]. We point out that the approach developed in the present manuscript is radically different, as it involves the Chern number of a non-degenerate and well-isolated dark band; in particular, the sum rule in Eq. (9) relates the dark band’s Chern number to the Chern numbers of the individual bare bands, without requiring the use of an open-system formalism."

Regarding the question "what makes HH bands reminiscent of the Landau levels?": HH bands can be (nearly) flat and simultaneously have a Chern number equal to one.

We believe that our revised manuscript takes all the suggestions of the Referee into account, and that it is now in a suitable form for publication.

List of changes:

- The abstract has been revised so as to better highlight the asset of our dark-state scheme, namely, the possibility of realizing nearly flat Bloch bands with *predictable* and tunable Chern number.

- Section 2 has been revised in view of providing a more pedagogical introduction to the Lambda setting; we also provide a more explicit definition of the Lambda system of Bloch bands in Section 3 [see Eqs. 7 and 8].

- The most important revision concerns the addition of the mathematical proof of the sum rule [Eq. 9], which constitutes the central result of our work; see Eqs. 9-12 and related text.

- A clear comparison with Panas2020 [Ref. 63] is provided at the end of Section 3.

- Section 5 presents new numerical simulations of the loading and detection scheme, which are more compatible with realistic experimental time scales; see Figure 3, Table 1 and related text.