Higher-order topology protected by latent crystalline symmetries

In this work, the authors study the consequences of latent symmetries on the topological classification of a lattice system. A latent symmetry is a symmetry of a reduced Hilbert space obtained by a projection called an isospectral reduction that is, importantly, not a symmetry of the extended Hilbert space (of course, if a symmetry is already preserved in the extended Hilbert space, then it must be also a symmetry of the reduced Hilbert space if the projection is faithful). The authors argue that latent symmetries are enough to induce higher-order topology in the restricted Hilbert space. An important aspect of the reduced Hamiltonian obtained after ISR is that it is energy dependent. So the reduced Hamiltonian is not just a mere effective Hamiltonian typically found in second-order perturbation theory where the energy E in (E − H)−1 is set to whatever the characteristic energy happens to be. The resulting eigenvalue problem is thus a nonlinear problem since in H(E)ψ = Eψ, the energy appears on both sides. The authors show how to construct various lattices that respect latent symmetries, and demonstrate that they host corner charges characteristic of HOTI’s. They also assess the robustness of these modes against various perturbations. Overall, I find the present study interesting and, as far as I can tell, technically sound. The draft is mostly clear, and the physics is generally logically presented. I believe that topological insulator community will appreciate this work. I therefore recommend this work for publication in SciPost pending the authors’ responses to the following comments.

Response: We thank the referee for finding our study “interesting”, the draft “clear”, and for recommending our manuscript for publication in SciPost. In the following, we reply to the comments of this referee.

1) If possible, I would like the authors to address the following issue more fully in the text. I believe the authors have already, in one way or another, hinted to this issue in the present draft, but, at least in my opinion, it is still a bit unclear. Unless I misunderstood the authors’ intent, it appears to me that in the topological classification of the system of interest, it is still necessary to have the full symmetry representation of the operator Q = Cn ⊕ Q, as shown in Sec. IV.A. The purpose of the latent symmetry Cn is to imply that Q exists, but the topological classification of the system still relies on knowing Q explicitly. Is this an accurate statement? In other words, I would appreciate an explanation as to whether it is possible to diagnose the topology of the restricted basis based solely on the latent symmetry and the restricted Hamiltonian h(k, E).

Response: The argument here should be the other way around. The existence of a latent symmetry directly yields the full operator Q from which, indeed, the topological characterization can be done. Without the latent symmetry, it would be difficult to explain why the matrix Q exists, and even more difficult to find it. The latent symmetry provides an elegant solution to this problem. Moreover, the topological classification is, contrary to the referee’s understanding, fully characterized by the latent symmetry Cn, rather than the full symmetry Q. The restricted Hamiltonian, whenever it features a topological character, is sufficient to understand the topological phase of the full system. We are sorry that these points were not expressed in a clearer way in the original manuscript.

Action taken: We have now made it clearer in the conclusions that one can characterize the topological phases of the entire system by studying only the latent symmetry. At the level of the full Hamiltonian, the topology is protected by a non-spatial discrete symmetry Q, which can be fully characterized by a spatial latent symmetry Cn.

2) In the discussion of the extended SSH model, the authors write ”Secondly, besides revealing latent symmetries, the ISR also allows for a simpler topological characterisation of Bloch Hamiltonians.” However, in that discussion, the authors only show that one can find gap closings. While gap closings typically accompany a change in topology, I do not think that gap closings always imply a topological phase transition. So, that discussion still leaves open the question as to whether the topology can be characterized by the reduced Hamiltonian.

Response: We agree with the referee that indeed, a gap closing by itself does not necessarily imply a topological phase transition. However, since these models reduce to models that are known to host topological phases, the gap closings can (and often will) yield topological phase transitions. Finding the gap serves as a first step towards resolving the topological phase diagram of the latent systems. One could say that these gap closing conditions give the phase boundaries in a phase diagrams. For example, for the latent SSH model, we found the gap closing energy to be equal to E∗ = ±√2w (end of section IIA). The known condition for topology in the SSH model, i.e. |w| = 1 can then directly be used to predict the topological phases of the latent model. Its generalization is |s(E∗, w)| = 1, with |s(E∗, w)| < 1 indicating a topological phase.

Action taken: The end of Section IIA has been further elaborated.

3) In the conclusion, the authors write, ”This reveals another strength of our method: if, under an isospectral reduction, a Hamiltonian reduces to an energy-dependent version of a known model, then properties of the known model can be used to characterize the full Hamiltonian.” The key here is reduction to a known Hamiltonian, then the topology can be inferred. What if the reduced Hamiltonian does not look like any known Hamiltonian, then can it be topologically diagnosed on the basis of the reduced Hamiltonian alone

Response: In this case the ISR still proves to be useful, as it reveals the symmetry that underlies the Hamiltonian. This then allows for a topological characterization using the appropriate invariants, for example symmetry indicators. In the manuscript, we investigated models that reduce to known models. Although this simplifies the calculations, it is however not a necessity. This point was already emphasized in the conclusions of the paper.

Action taken: The conclusions have been rewritten to emphasize this point even further. Furthermore, I suggest the following stylistic changes to the authors which, in my view, would improve the readability of the text. Response: We thank the referee for suggesting these changes, which we shall comment on in- dividually in the following. All of them have been incorporated into the revised version of the manuscript.

4) In the caption of Fig.1b, is the OBC band structure that of the topological phase only? If so, that should be noted in the caption. I suspect if the OBC band structure were for the trivial phase, it would not feature the edge states.

Response: The referee is correct that the OBC spectrum is calculated for the topological phase. The trivial phase would not show in-gap edge states. Action taken: The caption has been adjusted to state that the spectra are calculated for the topological phase.

5) I find this clause a bit confusing: ”associated with a crystalline symmetry dividing the lattice in n sectors, each spanning an angle of 2π/n rad”. Here, I suspect the authors mean to say that the filling anomaly is somehow mandated by/related to Cn symmetry. In the SSH model, n = 2. By the way I read this sentence, it seems to imply that there is some relationship between n and η. Is that true or am I misunderstanding something?

Response: Indeed, this is true. This relation was first introduced by Benalcazar et al. [1]. The symmetries of the unit cell in a system under open boundary conditions (OBC) impose constraints on the system’s behavior. In particular, when the system is in a topological phase, these symme- tries enforce an n-fold degeneracy of the corner states. The associated charge centers are arranged such that their full occupation by electrons requires either leaving all degenerate states empty or completely filled. If, instead, the electron filling is chosen to match that of the underlying periodic boundary condition (PBC) system, the electron must be distributed across all degenerate states. This distribution leads to fractionalized charge localized at the corners.

Action taken: The sentence has been reformulated to clarify the relationship between n and η.

[1] W.A. Benalcazar et al. Quantization of fractional corner charge in Cn-symmetric higher-order topological insulators, Phys. Rev. B 99, 245151 (2019).

6) What is the difference between the Wilson loop and the Zak phase? Perhaps one or two sentences explaining the difference and similarity between the two concepts would be helpful to the reader.

Response: When there are no band crossings, the Zak phase for a certain filling is equal to the sum of the Wilson loop eigenvalues for that filling. Consequently, they both describe the same topological features. For the sake of completeness, we discuss both in the text.

Action taken: The relation between the Zak phase and Wilson loop is now mentioned in the text.

7) Eq. (20) has no explanation for any of its symbols. What are Y, X, M and K? I infer that these are high-symmetry points in the BZ of a C2, C3, C4 or C6 system, but it would be good to say so, if true, explicitly in the text. The same issue appears with a1 and a2 being undefined in Eq. (23).

Response: We thank the referee for pointing out that the high-symmetry points in the BZ and the lattice vectors were not explained in the text. They have now been introduced correctly.

Action taken: Some sentences have been added to explain the meaning of X, Y, M and K and a1 and a2.

8) There is a minor grammatical error: ”Before including disorder, we show in Figs. 10(a) and 10(b) show the open boundary spectra of a triangular flake of the geometric and latent breathing Kagome lattice, respectively.” The second word ”show” should not be there.

Response: We thank the referee for pointing out this minor error.

Action taken: The grammatical error has been corrected.

In this manuscript the authors have applied the concept of latent symmetries to classify topological phases of lattices with distinct rotation symmetries (C2, C3, C4 and C6 topological crystalline insulators). Starting from the original Hamiltonian, the isospectral reduction (ISR) is applied in order to reduce the problem’s dimensionality. Interestingly, by doing this transformation a non-apparent (hidden) symmetry can, in fact, demonstrate a symmetry in its latent symmetry (the effective Hamiltonian). Supported by this argument, the authors have further demonstrated that the geometric rotation symmetry is not a required constraint for the appearance of topological phases. Indeed, the results presented non-trivial fractional corner charges in 2D lattices regardless holding the Cn symmetry. The results presented in this manuscript are sound, and I believe that is attractive to the readers of SciPost. Notice that while reading the manuscript, I found some few minor points that could be added/modified to improve the manuscript’s readability, which are described in the Requested Changes.

Response: We are delighted to read that the referee considers our results “sound” and “attractive to the readers of SciPost”. We are also grateful for their suggestions on improving the manuscript’s readability. All of them have been incorporated into the revised version of the manuscript. In the following, we adress these comments and indicate the corresponding changes that we made in the revised manuscript.

1) I would suggest authors to cite a reference for the SSH model in the beginning of Sec. II.

Action taken: A reference has been added.

2) While describing the Hamiltonian for each example (Eq.5, Eq. 25, Eq. 28, Eq. 34 and so on), I believe that stating the basis that the Hamiltonian is being written could make clearer.

Response: We agree that it can be instructive to explicitly show the basis numbering of the latent unit cells/lattices. In all our Hamiltonians, we take the red sites to be the first n entries of the Hamiltonian, the white sites follow after. For clarity, this has now been indicated in Figure 2. (previously Figure 3.)

Action taken: The basis numbering of the latent cells has now been indicated in Fig. 2.

3) While writing the Eq. 34, Eq. 40 and Eq. 46, the element H^{n=2,3,4}_L appears in the equations. I understand that they are previously described (whether in the main text or in the appendix), however, I would suggest to point to the equations where they first appear. Also, the symbol for null elements first appears in Eq. 40 without describing what is.

Action taken: (back-) references to the initial definitions have been added throughout the whole manuscript.

4) In Fig. 10 (b) the green dashed lines are marking the topological corner states. What about the characteristic of the other flat band at zero energy?

Response: The flat band at zero energy can be explained purely from an analysis of the unit cell, which we depict in Fig. 1(a) of this report again for convenience. For any value of t0, the cell supports three states that all have zero energy and vanish on the red sites {1, 2, 3}; these states are depicted graphically in Figs. 1(b) to (d). States with the characteristic of perfectly vanishing at some sites are known in the literature as “compact localized states” (see, for instance, the Review “Artificial flat band systems: from lattice models to experiments” by Leykam et al., Advances in Physics: X 1473052, 2018). Since these states all vanish on the sites {1, 2, 3} that connect the given unit cell to other unit cells in the lattice, it can be easily shown that the full lattice also features a zero energy eigenstate that is at least 3N-fold degenerate, with N being the number of unit cells. These 3N states can then be superposed to form Bloch states, and lead to three flat bands at zero energy.

Action taken: The above explanation of the three flatbands has been added to the appendix.

Attachment:

Latent symmetries offer the possibility to determine the physical properties of systems beyond the paradigm of conventional geometric symmetries. They are based on the isospectral reduction techniques in graph theory and provide an intricate link between the reduced effective system that exhibits a (hidden) symmetry and the original system. Cospectrality and Hamiltonian matrix power diagnosis underly this approach and make it constructive. Remarkably, the hidden symmetry on the reduced system can be extended to a full symmetry of the original system, which is however, in general, not of geometrical origin but a ‘complicated’ combination of the geometrical operation on the reduced system plus a e.g. general orthogonal transform of the remaining degrees of freedom. While latent symmetries have been used in previous works to e.g. design flat bands or explain degeneracies, the present work focuses on the extension of the concept of latent symmetries to higher order topological properties of crystalline lattices. This is a topic of immediate interest in the course of the recent developments moving from ‘traditional’ topological insulators to higher order ones, where higher order polarizations are quantized and lower dimensional edge states dominate the phenomenology. The present work represents major progress in terms of further developing latent symmetry concepts to topological band structure. It is a remarkable contribution to the state of the art of the field and should definitely be published in Scipost physics. Its merits stem from the fact that higher order symmetries i.e. multiple discrete rotational ones are conceptually implemented to represent latent symmetries and goes all the way to showing a series of concrete examples for lattices exhibiting arbitrary higher order topology by combining ‘simple’ models via so-called primitive generators. While this work is overall well-structured and designed, it is of comprehensive character and dense in information, which sometimes challenges the reader to follow. In the following I give a list of points where comprehensibility could be improved in order to improve on the accessibility of the work to a somewhat broader audience.

Response: We thank the referee for their kind words on the manuscript, in particular, for writing that the topic of the manuscript is of “immediate interest”, for considering it to be a “remarkable contribution”, and for recommending its publication in Scipost physics. We are also very grateful to the referee for their comments on improving the accessibility of our work. All of them were considered and most of them incorporated into the revised version of the manuscript. In the following, we go through these comments and indicate the corresponding changes that we made in the revised manuscript.

1) I would (this is a matter of taste) avoid stating that things are ’simple’,’trivial’ or ’easy’ but refer to being of ’immediate’ or ’straightforward’ character.

Response: We agree with the referee that these statements could be better phrased. In the revised manuscript, we avoided these statements.

Action taken: We replaced all sentences containing these adjectives with more neutral/formal ones.

2) On page 2, the mirror symmetry M of the SSH model is exposed, which is a geometrical symmetry of the model. Does chiral symmetry play any role in what comes later ?

Response: The usual topological characterization of the SSH model in terms of the AltlandZirnbauer classification indeed relies on the presence of chiral symmetry. The chiral symmetry allows for the definition of a winding number which captures the bulk topology of the system. However, the SSH model does allow for a second type of characterization, which is in terms of its crystalline symmetries, i.e. the mirror symmetry. This leads to symmetry indicators, as outlined in the manuscript. The SSH model is a special case, where the same topological phase can be characterized by two different approaches, based on different types of symmetry. For the higher-order two-dimensional phases this is not generally true, as chiral symmetry is often absent for these systems, and as a consequence, characterization in terms of symmetry indicators is the correct approach. For this reason, we did not investigate the role of chiral symmetry in the main text.

Action taken: A brief discussion on the role of chiral symmetry in the SSH has been added in the text.

3) Page 3, right column, first paragraph. The authors write: ”Secondly,....complicated structure.” This is a little vague, can you be more precise in your statement.

Response: It is unfortunate that the referee found this statement vague. The next paragraph following this sentence explains exactly the reason why it is complicated: there is no simple geometric interpretation of the symmetry.

Action taken: We have changed the formulation of these two sentences such that the statement appears less vague to the reader. The new formulation states directly that the symmetry does not have a direct geometrical interpretation.

4) Page 3, right column, second paragraph. The sentence ”We then build a lattice...between sites S.” is a bit hard to follow: the ’union of sites S’ makes it cryptic.

Response: We agree with the referee, in the revised version this sentence has been rephrased.

Action taken: The sentence has been changed to state that the ISR gets performed simultaneously on all sites in all unit cells.

5) Page 3, right column, below equation 11. ”The existence of Q is central....of this work.” Later on this is not exposed enough. Can you make this clearer why, and later on refer to it again.

Response: We thank the referee for raising this point. The existence of the matrix Q guarantees that to every latent symmetry there is a corresponding symmetry of the underlying full Hamiltonian. It is thus the central link between two worlds, namely, the isospectral reduction and the original setup. In particular, the matrix Q allows us to apply the machinery developed for the treatment of HOTIs with geometric symmetries also to latently symmetric HOTIs.

Action taken: In the revised manuscript, we have modified the passage right below equation (11) to clarify the importance of the existence of Q. We now also refer to its importance again in the beginning of Sec. III. B.

6) Page 4, left column, second paragraph. What do you imply by ”...a simpler topological characteriation...”. The passage followed by ”More concretely,...” is somewhat implicity and difficult to follow for the reader, if they had not read the previous latent ssh work. Can you improve on understandability here ?

Response: We imply that the topological characterization is converted to solving an algebraic problem instead of calculating a (multiband) topological invariant. However, the specified section was written in a dense way, which assumed some knowledge of the previous latent SSH paper. Especially the step from equation 12 to 13 were not directly obvious. In the revised manuscript, this paragraph has been rewritten to better explain how the latent symmetry simplifies the problem.

Action taken: The section has been rewritten to be more understandable.

7) Page 4, right column, first sentence of section ”B. Further perspectives...”. I think the authors can do better in formulating what they want to express. Try to be balanced.

8) I appreciate that the authors try to put things in a proper context thereby being explanatory. But e.g. in saying ”In this case, ..... symmetries as well.” this seems to be too general to me. The paragraph ”There are, nevertheless.....simplifying eigenvalue problem:” could be compactified in one or two sentences. A similar statement holds for the passage ”On the other hand....when embedding those unit cells into a lattice.” on page 5, left column.

Response: We are grateful to the referee for these comments (both of which relate to Sec. II. B), and agree with them.

Action taken: In line with the referee’s comments, we have shortened this section and also polished the line of argumentation.

9) Equation 20: Please explain notation (Brillouin zone points of high symmetry, Xi, Yi, M...).

Action taken: A comment was added on the meaning of Xi, Yi, M, etc.

10) The footnote 4 is not understandable.

Action taken: The footnote has been rewritten to provide an example. It is now footnote 6 because two new footnotes have been inserted before it.

11) Page 7, right column, second paragraph. The sentence ”By carefully analyzing....the latent Cn symmetry.” needs more explanation ! What is to be done, how?

Response: We thank the referee for this comment. We had omitted the details of this procedure, as we found them to be rather technical. In the revised version of the manuscript, we present them.

Action taken: The corresponding part in the manuscript has been expanded accordingly.

12) Page 9, left column. Statement ”...such that the lattice as a whole keeps the symmetry.” This is not precise enough. The lattice has a combination of translation plus internal unit-cell symmetry.

Action taken: We emphasize now in the text that both the internal degrees of freedom and the coupling between different cells, should respect the Cn symmetry.

13) The sentence ”....for clarity, we....is Cn symmetric.” is unclear.

Response: We regret that this sentence is unclear. The idea is that, when constructing a lattice of a finite set of unit cells, one has to perform the ISR to all those cells to obtain the latent symmetric reduced Hamiltonian. That is, each cell has a set of sites S (red sites) and the combination, or union, of all these sites S gives the total set of sites over which the ISR should be performed.

Action taken: The paragraph has been rewritten to better explain what is meant by the simultaneous reduction.

14) Page 9, right column. Paragraph below equation 31. Try to explain this a little better, it is clumsy.

Response: We regret to hear that the referee found this paragraph clumsy. However, without further specification, it is difficult to fully understand what the referee would like us to change. Nevertheless, we went over this paragraph again and attempted to rewrite it in a clearer way.

Action taken: The paragraph has been rewritten. It now follows Eq. (33) because two additional equations have been inserted previously.

15) Concerning figures. The (w,t) phase diagrams need some more explanation in the figure caption.

Response: Since w enters the Bloch Hamiltonian only as an identity term, it does not alter the band topology of the system, and the phase diagram does not depend on its value. We emphasized this in the revised version of the manuscript.

Action taken: The description of the w, t phase diagrams has been elaborated in the figure caption.

16) Page 10, right column, below equation 33. If you put w = 0 then you have independent SSH chains - that is the well-known standard SSH case. So, why addressing this at all?

Response: The referee is indeed correct that this limit decouples the chain into independent SSH models. This limit was chosen for two reasons: (1) As remarked in the response to point 15, the phase diagram does not depend on the value of w, making w = 0 the easiest case to study. (2) This limit was also taken for C2-symmetry in the initial paper by Benalcazar et al. [1], on non-latent rotational symmetry. In an analogy to their paper, we took the same limit.

Action taken: A footnote has been added discussing the w = 0 limit and how it relates to the w ̸= 0 case.

[1] W.A. Benalcazar et al. Quantization of fractional corner charge in Cn-symmetric higher-order topological insulators, Phys. Rev. B 99, 245151 (2019).

17) Page 12, right column (and several other places), equation 40. I would put the zero blocks as boldface with index, but not as striked out zero.

Response: We consider this point to be a matter of taste. Since in all our previous works and in the works of many other colleagues in the field striked-out zeros have been used, we prefer to keep the notation here for consistency.

Action taken: None. We kept the notation chosen previously.

18) Ref. 21 should be replaced/updated. Page 1, ref.16 citation should be replaced by a more immediately accessible one.

Action taken: Reference 21 has been updated (it is now reference 22). Reference 16 has been updated to include a web link to the pdf version.

19) The conclusions are not well written, and not well structured in view of the content of the manuscript. I suggest to carefully think over how to do it, and then write it.

Response: We regret to hear that the referee thinks the conclusions are not well written or structured in view of the content. After implementing all other suggestions, we went over the conclusions again and rewrote them.

Action taken: The conclusions have been fully rewritten.