Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker

Errors in user-supplied markup (flagged; corrections coming soon)

The thank the referees for their insightful comments. In the following, we reply by directly quoting the referee reports. Their original text is given between dashes (---).

---
Anonymous Report 2 on 2022-10-28 (Invited Report)
---
We thank the referee for their extensive comments and criticisms which have really helped us improve our work.

---
Strengths
1. This work introduces generalized physical quantities derived from the so-called "local Chern marker" (introduced by Bianco and Resta [7]), which could be used to signal phase transitions in Chern insulators.
2. These generalizations have the potential to introduce novel probes and concepts in the field of topological quantum matter.
Weaknesses
Most of the introduced markers are equivalent to well-known quantities (e.g. the nonlocal Chern marker in Eq. 38 is equivalent to the Wannier-state correlation function of Ref. 25;
---
Firstly, the nonlocal Chern marker itself is a new concept proposed in our manuscript. We are unaware of any article that has claimed that the off-diagonal element of the Chern operator can be used to diagnose topological phase transitions. We realized that the reason that the nonlocal Chern marker becomes more and more long ranged as the system approaches the topological phase transition is because it is equivalent to the Wannier state correlation function. In fact, this equivalence implies a remarkable feature, namely the Wannier state correlation function can be simply read off from the off-diagonal elements of the Chern operator constructed from diagonalizing the lattice Hamiltonian, which is a very simple numerical recipe that further advances the theory of topological phase transitions. Motivated by the referee’s criticism, we have emphasized this new understanding about Wannier state correlation function after Eq. (33).

---
The "generalized nonzero temperature local Chern marker" in Eq. 23 seems to simply correspond to the local Hall response at finite temperature). In this sense, it is not clear whether these "generalized markers" are in fact useful/novel.
---
The referee correctly pointed out this equivalence between the finite temperature local Chern marker and finite temperature local Hall response. In the revised manuscript, our emphasis is on the issue of measuring the Chern marker: The global Hall response is certainly the quantized Hall conductance of the whole 2D Hall bar, but there is no clear experimental protocol to measure the local Hall response in a realistic finite temperature system. The present work provides a concrete answer to this question, and clarifies that the local Chern marker can be measured by atomic scale thermal probes as the local heating rate caused by circularly polarized light (please see reply to referee’s other question below). In addition, our work suggests that the global Hall conductance can be measured from the opacity of 2D systems to circularly polarized light, similar to those measured in graphene in Ref.~28 and 29. This clarification therefore presents a clear experimental path to the measurement of the Chern number and Chern marker, as now emphasized in the introduction, Sec. 2.2, 2.3, and conclusions.

---
Related to the point above: the experimental access to these "generalized markers" is not properly described in the manuscript. What are the probing fields and detection tools that are required to access these markers in experiments?
---
Indeed, in the previous version of the manuscript, the concrete measurements of the Chern number and Chern marker were not properly described. In the revised manuscript, we elaborate that the finite temperature Chern number can be measured from the opacity caused by circularly polarized light, which we explain in detail below. Note that the opacity caused by linearly polarized light has been measured in 2D materials for more than a decade since the discovery of graphene, and changing from linear to circular polarization should be an easy task, pointing to the feasibility of our proposal.

On the other hand, the local Chern marker can be measured by atomic scale thermal probes, such as the scanning thermal microscope in Ref. 37 ~ 40, as the local heating rate against circularly polarized light. This proposal indicates that these atomic scale thermal probes can be used to examine how the chemical and structural defects, which always occur in realistic 2D materials, affect the topological order locally. This important insight is now emphasized at the end of Sec. 2.3 and conclusions.

---
In general, the information provided by these generalized markers (such as the "topological correlations" associated with the "Chern correlator") remain vague.
---
After reflecting on the referee’s criticism, we agree that to interpret the magnitude of Chern correlator as a kind of topological correlation can be misleading, as it is not very clear what a topological correlation would really mean. What we observe is that the Chern correlator, whose spatial integration gives the integer-valued Chern marker, has a spatial profile that fluctuates more drastically as the system enters the topologically nontrivial phase, and it corresponds to a nonlocal current-current correlator that is a physically meaningful quantity. Therefore, a more appropriate articulation is to interpret the magnitude of Chern correlator as a kind of internal fluctuation of the Chern marker, which becomes more dramatic in the topologically nontrivial phase, indicating that the nonlocal current-current correlator will also manifest significant spatial fluctuation. We have followed the referee’s suggestion to remove the word “topological correlation” everywhere and emphasize this fluctuation picture in the introduction, Sec. 4.1, and conclusions.

---
The manuscript contains a series of vague and/or misleading statements.
---
We reply to the referee’s suggested vague and misleading statements below.

---
Report
This work builds on the concept of "local Chern marker" (introduced by Bianco and Resta [7]) to propose a series of generalized physical quantities, which could be used to signal topological phase transitions in Chern insulators. While this work makes interesting links between different quantities (markers, response functions, correlation functions, ... ), it is not clear whether the proposed markers provide any new, concrete, or relevant information on topological states and their phase transitions. In this sense, I find the scope of this work rather limited.
---
Following the referee’s comment, we have substantially rewritten the manuscript to clarify the new aspects introduced in our work and their role in direct experimental observations of bulk topology, which we briefly summarized below.

(1) We propose that the Chern number can be simply measured from the opacity of 2D materials.

(2) We suggest that Chern marker can be measured by scanning thermal microscope as the local heating rate.

(3) We propose a Chern correlator to quantify the internal fluctuation of the Chern marker.

(4) We introduce the nonlocal Chern marker to detect topological phase transition in real space, which can also be used to extract the Wannier state correlation function.

More details of these points are given in the reply to referee’s comments below.

---
- In the introduction, I found the following sentence misleading: "The bulk band structure, on the other hand, shows the same gapped energy spectrum in both the topologically trivial and nontrivial phases, and therefore bulk measurements not involving edge states are often not considered feasible to identify the Chern number". This statement seems to contradict the well-known TKNN result, according to which the Hall conductivity (as obtained from Kubo's formula, considering a system with periodic boundary conditions) is related to the Chern number. From the TKNN result, it is clear that the Chern number *can be identified* from "bulk measurements not involving edge states", which is in sharp contrast with the authors' statement. In this sense, I believe that this statement should be revised.
---
We thank the referee for highlighting this confusing statement. What we intended to convey in this sentence is that bulk “experimental” measurement of the TKNN invariant, i.e., the measurement of quantized Hall conductance, relies on the existence of edge state (one should keep in mind that the quantized Hall conductance in reality is given by the number of conducting edge states at the boundary of the Hall bar). Thus, for systems without a boundary, such as a torus, Hall conductance measurement would not be feasible because there is no edge state, and one must seek for other means to measure the TKNN invariant. Equivalently, one may also ask how the TKNN invariant can be measured deep inside the bulk without using transport measurements. These points motivated us to propose that one can measure the spectral function of the opacity and then integrate it over frequency, which is an optical technique entirely independent from whether the edge state occurs or not. Nevertheless, we do agree with the referee that this sentence is misleading, and have revised it in the new submission.

---
- Page 3: the sentence "To relate the Chern number to our proposed Chern marker" introduces an ambiguity: here the authors seem to refer to the well-known Bianco-Resta marker, and not to *their* new markers (introduced later in Section 4). Similarly, a clear citation to Bianco and Resta [7] is needed above Eqs. 9 and 12. [Note: Eq. 12 should end with a full stop '.'].
---
We have cited the Bianco-Resta paper in the Chern marker formula in the new version in Eq. (10). Our finite temperature generalization of Chern marker is now given in Sec. 2.3.

---
- I was puzzled by the opening of Section 2.3 "The result linking the Chern number to the circular dichroism can be generalized to nonzero temperatures using linear response theory", which seems to announce the derivation of a fundamentally new result. As far as I can see, it is trivial to generalize Eqs. 15-16 to finite temperature, by simply inserting a Boltzmann weight [exp (- E_n/kT)] inside the sum of Eq. 15. By doing so, the right-hand side of Eq. (16) becomes the finite-temperature Hall conductivity, which (as far as I can see) is equivalent to the quantity displayed in Eq. (23) and Fig. 1(d). Similarly, the quantity shown in Figs. 1 (a)-(c) seems to correspond to the differential rate [the integrand in Eq. 16] at finite temperature, which is simply obtained by inserting the Boltzmann weight in Eq. 15. In this sense, I could not identify the novelty (or non-triviality) here.
---
Taking into account the referee’s criticism and the comments from other researchers in the field, we have decided to completely abandon the real space linear response theory in the previous version that intends to relate the Chern marker to charge polarization. In fact, there is some ambiguity in both the theoretical interpretation and numerical calculation of the previously presented formalism. In the revised version, we take the route of momentum space formalism based on the fact that the integration of Berry curvature gives the quantized Hall conductance. Within this context, the optical Hall conductivity at zero temperature has been discussed previously in Ref. 27. Our aim is then to first generalize this optical Hall conductance to finite temperature, and then suggest that it can be simply measured from the opacity of the material. This is why the paper has been largely modified.

We agree with the referee that the finite temperature formalism of optical conductivity can be easily found in any many-body textbook, so we do not take any credit for this. Rather, the new insight we brought in is the derivation of Chern number spectral function from the optical conductivity, and how it can be measured from the opacity of the material, as now given in Sec. 2.2

---
- About the quantity displayed in Eq. 23, the authors wrote "To summarize, Cd(r) is a real space generalization of a topological probe which is intimately related to the measurable charge polarization susceptibility". I found this statement misleading: indeed, it seems to me that the quantity in Eq. 23 simply corresponds to the local Hall response of the system at finite temperature. Altogether, I could not identify the novelty of the results/quantities presented in that Section 2.3.
---
In the revised manuscript, we have abandoned the real space charge polarization formalism, and follow the conventional momentum space Hall conductance derivation. This implies that the referee’s comment is correct, namely the Chern marker spectral function we have derived is physically equivalent to the local optical Hall conductance. Nevertheless, in Sec. 2.2 and 2.3 of the revised manuscript, we have brought in several new insights into this conventional formalism, including:

(1) Correctly formulating the Chern marker at finite temperature. It turns out that this is a highly nontrivial question. We find that to simultaneously demand that the Chern marker has to spatially integrate to the Chern number and include the effect of thermal broadening, there is only one way to define the Chern marker, as given in Eq (19) and (20). This point is elaborated more in the reply to referee’s question below.

(2) We link the Chern marker spectral function to local heating rate, implying that it is possible to measure it by atomic scale thermal probes. Note that although the local Hall conductivity has been investigated previously, we are unaware of any concrete experiment that can directly measure it. In contrast, the present work suggests to measure the spectral function by scanning thermal microscope and then integrate it over frequency to obtain the Chern marker.

---
- Similarly, I found the following sentence misleading (also below Eq. 23): "The generalization to nonzero temperature is an important milestone to address topology in more realistic scenarios". The notion of finite-temperature Hall response is well known, and in this sense, I believe that this is not a "new generalization" nor "an important milestone".
---
Indeed, we agree with the referee that this statement is exaggerated, so we have rephrased it to put the emphasis on the proposed opacity measurement. This serves as a concrete strategy to detect the topological order in realistic inhomogeneous systems at finite temperature. This way we also emphasize that the novelty of the manuscript is to present a bulk optical measurement protocol, and to convince the reader that such an experiment is entirely feasible according to the seminal opacity experiment in graphene.

---
- I would invite the authors to explain why the alternative expressions in Eq. 19-20 are in fact useful.
---
After adopting the momentum space optical Hall conductivity formalism in the revised version, we realized that Eqs. (19) and (20) are not alternative expressions, but the only expression that can define a Chern marker at finite temperature. The reasoning is given before Eq. (19), and briefly summarized below.

In the Chern marker formalism proposed by Bianco and Resta in Ref 7, it is known that an extra projector to the valence band states P has to be included to get correct numerical results. However, at finite temperature, thermal broadening renders the notion of valence band rather ambiguous. The question is then how one can consistently include an extra projector at finite temperature to produce a local Chern marker that spatially integrates to the global Chern number? We find that Eqs. (19) and (20) are really the only way that can satisfy all these requirements. Physically, it implies that the PxQ operator in the Bianco-Resta formalism in Eq. (8) has to be replaced by the X operator in Eq. (19) at finite temperature, a new insight that we have brought to the theory of Chern marker, which allows it to be interpreted as the local heating rate.

---
- Below Eq. (29), I found the statement "represents the shift to the neighbouring sites" unclear. Besides, a sketch of this model would have been useful.
---
Indeed the “shift” is not a good articulation. We now simply refer to {a,b} as the lattice constants in the two planar directions. Due to the multi-orbital nature and complex directional phases involving these, we think that a sketch may actually confuse the readers. Nevertheless, we have improved the description by first introducing the well-known momentum space Hamiltonian, and then explain that the corresponding lattice model can be obtained from a straightforward Fourier transform.

---
- Below Eq. (32): I found the following statement misleading "and tune the mass term M and temperature kBT to examine different topological phases", which seems to suggest (implicitly) that varying temperature could lead to different topological phases.
---
We agree that this sentence is misleading. We have changed it to “we tune the mass term M to examine different topological phases, and the behavior of this model at finite temperature T.”

---
- Above Eq. (31): The sentence "The Chern number and Chern marker spectral functions are equivalent in this homogeneous model, giving ... ", together with the following Eq. 33, seems to suggest that the Chern number is a well-defined topological invariant at finite temperature, which is misleading.
---
We thank the referee to reading our text in such detail. We have removed the Chern number part in this sentence and now just comment that “The finite temperature Chern marker spectral functions is a constant of r in this homogeneous model” as given by Eq 33.

---
- Below Eq. (32): The authors wrote "This Chern correlator represents a measure of topological correlation in the system" and then later "the Chern correlator (...) can therefore act as a proxy for the amount of spatial topological correlation in the system". Could the authors *rigorously define* the concept of "topological correlation"? What is topological about these correlations? Why should we care about these correlations (what new/specific informations do they provide)?
---
As mentioned in the reply to referee’s question above, we agree with the referee regarding the interpretation of the Chern correlator as a kind of topological correlation. We have now removed this statement everywhere in the manuscript. Rather, the Chern correlator is now interpreted as an internal fluctuation of the Chern marker, i.e., despite the fact that the Chern marker remains quantized, its internal structure varies with parameters, and is found to fluctuate more dramatically in the topologically nontrivial phase.

However, we also emphasize that it is possible to introduce the notion of topological correlation, not from the Chern correlator but from the nonlocal Chern maker in Sec. 4.2. The interpretation of the nonlocal Chern marker as a topological correlation is legitimate because it indeed has the physical meaning as a correlation function that measures the overlap between two Wannier states that are a certain distance away, and moreover it decays with a correlation length that diverges at topological phase transitions. All of these are in accordance with the usual definition of correlation function in statistical mechanics. In fact, one can even extract critical exponents and assign universality classes from this correlation function, as have been demonstrated in Ref. 30 ~ 32. We have remarked on this topological correlation interpretation of the nonlocal Chern marker in Sec. 4.2.

---
- About Fig. 4, the authors wrote "Note that the spectral function exhibits a rich spatial structure, indicating that the correlation within the system can vary a lot depending on the probed frequency". I had difficulties identifying any concrete information that one could get from that result (and the related statement in the text). What do we learn from this rich spatial structure?
---
The spatial profile in Fig 4, presented for different frequencies, has the physical meaning of the nonlocal current-current operator in the optical Hall response, i.e., what the current is produced at r by applying a field at r’. What Fig 4 shows is that the magnitude of this nonlocal current at r varies a lot depending on the frequency of the light at r’. Essentially this is because in the optical absorption process: light at a specific frequency excites electrons at a specific energy, and the wave functions of these electrons at site r contribute a particular pattern of the current at r. We now clearly discuss this physical picture at the end of Sec. 4.1.

---
- In the concluding paragraph, the authors wrote "All of these quantities can be measured in appropriately tailored circular dichroism experiments." I found that statement very vague.
---
Below we summarize the measurement protocol for the three quantities we introduced in this manuscript, namely the Chern number spectral function, Chern marker spectral function, and nonlocal Chern marker, which have been emphasized in the corresponding sections.

(1) We realized recently that the proposed Chern number spectral function can actually be measured in a very simple manner from the opacity of the 2D system to circularly polarized light, as now elaborated in Sec 2.2 and 2.3. This proposal is inspired by the seminal experiment on graphene in Ref. 28 and 29, which shows that the opacity of graphene is determined by the fine-structure constant. Our new insight is that if one shines a circularly polarized light on the 2D system under question, then the optical Hall conductance multiplied by the square of the electric field of the light gives exactly the absorption power of the system. The opacity is then given by dividing this absorption power by the incident power of the circularly polarized light. As a result, one can obtain the Chern number spectral function by simply subtracting the opacity at the two circular polarizations and then dividing by frequency, which is a very simple measurement protocol. In fact, it just corresponds to replacing the unpolarized light in the aforementioned graphene experiment by circularly polarized light. The Chern number can then be obtained by integrating this spectral function over frequency. This leads to a concrete experimental proposal to extract the Chern number, and can be used to examine the frequency sum rule proposed in Ref 27.

(2) The same reasoning also applies to the measurement of the Chern marker spectral function, which corresponds to the local absorption power of the unit cell at site r. Although the diffraction limit likely hinders the detection of local opacity in the atomic scale, we predict that this absorption power can be measured by atomic scale thermal probes as the heating rate against circularly polarized light.

(3) For the nonlocal Chern marker, because it is equivalently the Fourier transform of the Berry curvature, in principle one can first measure the Berry curvature in momentum space by some means, such as the pump-probe experiment proposed in Ref. 26, and then Fourier transform it to obtain the nonlocal marker. Moreover, the Berry curvature generally takes a Lorentzian shape near the gap-closing momentum, as have been pointed by many previous works, and the width of the Lorentzian is precisely the inverse of the decay length of the nonlocal marker. As a result, one only need to measure a very small region near the gap-closing momentum to know how the nonlocal Chern marker decays in real space, which is entirely feasible by pump-probe. In fact, a similar technique has been applied to investigate a small momentum region near the Dirac point of graphene in Ref. 54.

---
- In general: It would be useful to clarify (i) what are the physical quantities that are genuinely new in this work, (ii) how they could be measured in practice (what probing field, what detection technique is needed?), and (iii) what new informations/advantages do these new quantities offer (as compared to well-known quantities such as Hall responses, dichroic signals, Fourier-transformed Berry curvatures, ... ).
---
Below we list the quantities that are genuinely new in our work, how to detect them, and what information they can provide:

(1) The Chern number spectral function - this spectral function can be extracted from the opacity of 2D materials against circularly polarized light at finite temperature, measured in units of fine structure constant just like that has been done in graphene, and after a frequency integration yields the Chern number. This offers a very simple and ubiquitous experimental protocol to extract the Chern number for any 2D material. In fact, the experiment in graphene strongly suggests the feasibility of this experiment in practice.

(2) The Chern marker spectral function - we recognize this spectral function as the local heating rate against circularly polarized light, which may be detected by atomic scale thermal probes like scanning thermal microscope. The frequency integration of this spectral function gives the Chern marker. The advantage of this method is that is allows to probe the influence of spatial inhomogeneity, such as disorder and grain boundary that occur in realistic materials, on the topological order.

(3) The Chern correlator - this correlator is extracted from the nonlocal current-current correlator in our linear response theory, and represents the internal fluctuation of the Chern marker. Although we do not recognize an experimental protocol that can directly measure this nonlocal response, this Chern correlator serves as a theoretical tool to quantify the internal fluctuation of the Chern marker which we found to be larger in topologically nontrivial phase.

(4) The nonlocal Chern marker - this quantity can be obtained from a Fourier transform of the valence band Berry curvature measured in momentum space, for instance by the pump-probe experiment proposed in Ref.~26. Because this nonlocal marker is equivalently a Wannier state correlation function, it also implies that this correlation function can be simply evaluated by diagonalizing the lattice Hamiltonian, and can be used to detect topological phase transitions directly in real space.

---
Anonymous Report 1 on 2022-9-10 (Invited Report)
---
We thank the referee for their comments and questions. We have addressed them in the revised version of the paper.

---
Strengths
1- The paper provides a useful conceptual extensions of the Chern marker formalism, demonstrating that it can be recast in terms of response functions of the system to circularly polarized light. Specifically, the authors define a Chern marker correlator, a Chern marker spectral function, and a non-local generalization of the Chern marker
2- The interpretation of these quantities as response functions provides a straightforward generalization of the quantities to finite temperature.
3- The proposed quantities are experimentally measurable by probing the response to circularly polarized light.
Weaknesses
1- The data presented in the paper show a few numerically evaluated snapshots of the new quantities, but fall short of a systematic study. As a result, the paper cannot make strong conclusions about the general usefulness of these concepts.
Report
This is a well written paper, which introduces new concepts in the study of real-space Chern markers.
The work provides new tools to investigate topological bands, which I believe satisfies an essential criterion for this journal, to "Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work".
The potential for an experimental measurement of the proposed quantities is also promising for follow-up work.
I recommend the paper for publication, subject to minor changes. The authors may with to improve the paper by including a more systematic discussion of the different Chern response functions in different regimes.
Requested changes
1- The Chern insulator model given in equation (29) looks like a version of the two-orbital square lattice model known in the literature on (fractional) Chern insulators, corresponding to the spin polarized version of the model introduced in B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006). An original reference should be included.
---
To incorporate referee’s suggestion, we remark that we have recently generalized the formalism in the present paper to propose a spin Chern marker for 2D time-reversal invariant systems in Ref 55, and have used the Bernevig-Highes-Zhang (BHZ) model as an example in that paper. Remarkably, the behavior of spin Chern marker in the BHZ model is exactly the same as that of the Chern marker in the Chern insulators presented in this work. This is essentially due to the fact that the BHZ model is simply two copies of Chern insulators, one for each spin. Thus, the experimental protocols and nonlocal Chern marker in this manuscript can be straightforwardly generalized to the spinful version for time-reversal symmetric systems, which renders much broader applications. We have mentioned these points in the conclusions.

---
2- The authors note on page 13 that "the critical exponent of the correlation length ξ ∼ |M|−ν has been calculated previously from the divergence of Berry curvature". Given that the current paper is about Chern markers, it seems appropriate to also mention that the correlation length exponent has indeed been measured from the behaviour of the Chern marker also, e.g. in Ref. 46.
---
We thank the referee for this suggestion. We have added a sentence to highlight this point with the relative citation.

3- Consider adding more systematic exploration of the different Chern response functions.
---
In the revised version, we have added a new Fig 2 that shows the evolution of the proposed Chern number spectral function across a topological phase transition. The purpose of this figure is to show that as the system approaches a critical point, the spectral weight (which acts like the density of states of the Chern number) gradually shifts to lower frequency, and the low frequency part flips sign across the transition, consistent with the flipping of Berry curvature at high symmetry points in momentum space across the transition. On the other hand, at finite temperature, thermal broadening will reduce the spectral weight at low frequency, resulting in deviations from the quantized Hall conductance values. As a result, the sharp jump of the Chern number across topological phase transitions is smeared out at finite temperature. All these features should be readily measurable by the newly proposed opacity measurement in Sec. 2.2.

The figures in the paper now collectively demonstrate qualitative features that should be generic to all topological phase transitions, irrespective of particular parameter values. For instance, the Chern insulator we presented in Sec. 3 has three critical points as shown in Fig 1 (b). However, all of them have the same low energy spectral weight feature as shown in Fig 2, and the same decaying nonlocal Chern marker as shown in Fig 5. This point is now mentioned at the end of Sec. 4.2.