Real-space Formalism for the Euler Class and Fragile Topology in Quasicrystals and Amorphous Lattices

1. The question addressed in the manuscript: develop a local topological marker to describe “fragile” topological systems is both timely and interesting.

2. The clarity of exposition is of a good standard.

3. The authors provide a number of applications (and checks) of the proposed formalism. In particular, the authors apply it to a quasicrystal where the conventional method (computing Euler invariant in k space) is not directly applicable due to lack of translational invariance.

1. The manuscript in my opinion is lacking a sufficiently careful and clear development of the formalism.

In this manuscript, a real-space invariant to diagnose fragile topological states is proposed. The derived expression is used to investigate amorphous and quasicrystalline systems, both lacking translational invariance.

I think the topic is a great idea! I think the work should be published. I would like to provide some feedback, however. This is in the next field.

1.
The operator involved entering (for instance) equation (6) is P[UPU^dagger, VPV^dagger]. This operator also enters the Chern marker. One can see that it is directly related to the field strength matrix when there is translational invariance. So to me it seems to be a fairly direct *observation* — to obtain the Euler number you just need to replace Trace —> Pfaffian.

I think the readability might be improved if this is emphasized. It is fairly direct.

2.

What’s less clear to me is how the required symmetries translate to the real space expressions. As I understand, the integrated expression curl A_{12} is topological only when the u(k) can be taken to be real. Indeed, this makes the field strength matrix skew symmetric which is needed to even define the Pfaffian.

Please clarify:
How does this all translate to the real space expressions?
What are the required symmetries of the Hamiltonians considered and what does this imply about P[UPU^dagger, VPV^dagger]?

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

1. The authors have proposed a real-space Euler marker to characterize the topology of the Euler class. Such a formalism is very important as it can be used to identify the Euler topology in non-crystalline systems, such as quasicrystals and amorphous systems.

2. The paper is well written and organized.

1. The authors do not specify what |r> denotes in Eq. (7). This state is actually the Wannier function. However, when I first read it, I thought that it would the position state. Hope that the authors can provide the definition immediately following Eq. (7) for a better readability.

2. I wonder whether the submatrix defined over the Wannier basis in r-space is still antisymmetric in the absence of spatial translational symmetry. We know that the Pfaffian is defined only for antisymmetric matrices. In addition, how is the order of |W_1(R)> and |W_2(R)> determined in the calculations? This is important as the signs of the Pfaffian for distinct order are opposite.

3. I think Fig. S3 and Fig. 1(d) should use logarithmic coordinates to better illustrate the power-law decay of (1-|e|) versus L.

In the manuscript, the authors propose a real-space formalism of the topological Euler class by constructing a local Euler marker. Based on such a novel marker, the Euler topology in disordered systems, quasicrystals and amorphous systems is identified. In particular, the Euler topology supporting eight corner modes is found in a quasicrystal. I thus think that the results are very interesting and important, making a significant advancement in the field. I thus can recommend the paper for publication in SciPost Physics after some minor revisions.

1. Please provide the specific relation between $t_{\mu\nu}$ and ($V_{dd\sigma}$, V_{pp\pi}, $\dots$).
2. Please provide the energy spectrum under PBCs and OBCs, and the spatial distribution of nontrivial edge states for a specific $\Delta$ in region I of Fig. 1(b). The information can be compared with Fig. 3 to illustrate the upward shift of eigenenergies of corner states.
3. Please correct the citation “appendix ??” in line 188.
4. “$a_E/1-|e|$” should be written as $a_E/(q-|e|)$ in line 608 and Fig. S2.
5. The disorder parameter of the amorphous lattice in Fig. 3 is missing. Please add the information.
6. The construction methods of the amorphous lattice in Fig. S6(a) as mentioned in line 539 and line 661 are completely different. Please double check it.

Ask for minor revision

- very interesting subject and questions are addressed
- written clearly

- scientifically it does not solve the question, albeit interesting, that was posed
- refers to code online that in fact misses the subroutine that would is at heart of solving the issue
- claims made are not supported or only addressed in limits that were known

The manuscript by Li et al. proposes a real space formalism for describing fragile topology in tight-binding models of topological phases on amorphous lattices and in quasicrystals. In particular, the authors focus on topology captured by the Euler class. An analogue of the Bott index, i.e. a real space indicator known to reflect the Chern topology, is proposed for Euler topology. Furthermore, an attempt to formulate real-space topological Euler markers is provided. The proposals are followed by numerical results for disordered systems, which in clean limit host a fragile topological invariant, supporting the previously obtained results of other groups discovering topological phase transitions in disordered fragile topological systems. Finally, it is argued that the proposed real space indicators can indicate fragile topology in quasicrystals, where the standard symmetries ensuring a reality conditions, such as spatiotemporal inversion, are not present.

The problem of capturing fragile and Euler topology in real space is interesting and timely for the field. As such the questions asked are certainly interesting and highly relevant. However, as argued point-by-point below, one should strongly doubt that the introduced derivation of an appropriate Euler marker in the manuscript is sound. We thus think that the work does not support the claims made by the authors. Several important clarifications and improvements are therefore required. In essence, we note that the definitions only make sense when translational symmetry and proper momentum can be assigned. In contrast, when given a disorder real space system without knowledge of the momentum space Hamiltonian it could adiabatically connect to, the markers are not show/derived to be robust quantities to consider. We detail this in the following.

- The main criticism is that characteristic classes, such as Euler or Chern, mathematically describe vector bundles. Euler class characterizes real vector bundles of Bloch states, where the base space of the vector bundle is $k$-space; more precisely the BZ torus. Proposing a definition of the $r$-space Euler marker, which by the argumentation provided by the authors appears to be $a~priori$ a definition and is not really a derivation (contrary to the Chern marker for Chern insulators, derived by Bianco and Resta~[Phys. Rev. B 84, 241106(R)]), appears to not reflect the topology of the Bloch vector bundle defining the Euler class as a characteristic class, in general.

- The reason why this is a severe issue, is that the topological quantization of the invariant stems from the non-triviality of the (k-space) vector bundle of the Bloch states, which, in particular, when non-trivial (corresponding to the non-zero Euler class), manifests its topology by admitting no non-vanishing smooth sections in the Bloch bundle, etc. Within the r-space formulation considered here, there is no rigorous analytical argument for the topological quantization of, or a derivation-based $exact$ correspondence of the proposed Euler marker to, the Euler class invariant in the studied amorphous, disordered, or quasicrystalline systems.

- Very precisely, starting from the real-space Hamiltonian of a generic inhomogeneous system that satisfies the reality condition (e.g. preserving a C$_2$T symmetry), the authors failed to provide the exact expression of a topological invariant readily expressed in terms of the real-space eigenvectors such that it does not rely on the known clean reciprocal limit from which the Pf$_\text{occ}$ operation is defined (disentangling the “internal space", and defining the “hybridization" mentioned by the authors).

- This severe issue of the impossibility of evaluation of the Pf$_\text{occ}$ without a reference to the clean bands $\ket{u_{1\textbf{k}}}$, $\ket{u_{2\textbf{k}}}$, appears to be swept under the rug in the code provided by the authors (i.e checking the gitthub).

- Namely, where the subtle Pfaffian Pf$_\text{occ}$ is expected to be defined, the authors use a mysterious (and most importantly, not included in the code provided!) imported function (“import pf"), which in such state cannot be accepted as a valid and transparent method of computing the Euler marker. Indeed, the code provided only calculates the commutator, while the whole issue resolves around how to define the pfaffian WITHOUT referring to the k space state disentangling.

-As another remark, we note that the general theory that outlines multi-gap and Euler as well as fragile invariants, Physical Review B 102 (11), 115135 (2020), was not cited.

- finally I would advise to change the title as there is no formalism, neither is there explicit quasiperiodicity studied

Therefore, I cannot recommend the publication of the manuscript in its current form and hence require severe improvements.

see report

Ask for major revision