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 $(E-\mathcal{H})^{-1}$ is set to whatever the characteristic energy happens to be. The resulting eigenvalue problem is thus a nonlinear problem since in $\mathcal{H}(E) \psi = E \psi$, 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.

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 $Q = C_n \oplus \bar{Q},$ as shown in Sec. IV.A. The purpose of the latent symmetry $C_n$ 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 $\mathfrak{h}(k,E)$. 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. 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?

Furthermore, I suggest the following stylistic changes to the authors which, in my view, would improve the readability of the text.

(1) 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.

(2) I find this clause a bit confusing: "associated with a crystalline symmetry dividing the lattice in $n$ sectors, each spanning an angle of $2\pi/n$ rad". Here, I suspect the authors mean to say that the filling anomaly is somehow mandated by/related to $C_n$ 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 $\eta$. Is that true or am I misunderstanding something?

(3) 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.

(4) 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 $C_2,$ $C_3,$ $C_4,$ or $C_6$ system, but it would be good to say so, if true, explicitly in the text. The same issue appears with $\mathbf{a}_1$ and $\mathbf{a}_2$ being undefined in Eq. (23).

(5) 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.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1 - Well written;
2 - Describes with details the main points being discussed ;
3 - Nice amount of examples along the manuscript.

Some few parts of the manuscript could be better explained for a better readability. I believe that this flaw would be expected, as the it is a "long" manuscript and sometimes if normal to restrain some information.

In this manuscript the authors have applied the concept of latent symmetries to classify topological phases of lattices with distinct rotation symmetries (C_2, C_3, C_4 and C_6 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 C_n 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.

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

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.

3 - While writing the Eq. 34, Eq. 40 and Eq. 46, the element H_{L}^{n=2,3,4} 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.

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?

Ask for minor revision

Breaks new ground concerning the development of concepts and their application w.r.t. latent symmetries to topological crystals of higher order. Combines latest developments in the field of higher order topology with latent symmetry concept. Shows concrete examples and a very thorough analysis of the band structure and its topological character.

Occasionally a little difficult to read and follow due to many aspects being illuminated.

See the attached report.

See the attached report.

Ask for minor revision