Real-space approach for the Euler class and fragile topology in quasicrystals and amorphous lattices

I thank the authors for addressing my points. I have also read carefully the correspondence with Referee 2 who raises very valid points and has gone deeply into details and has valid points.

I am comfortable with recommending publication at this stage. The authors have proposed and motivated a topological marker and demonstrated its utility in a number of contexts. Future work may put the marker on a firmer theoretical foundation.

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

- Interesting topic
- Interesting and timely question

- No proof of actual method, rather vague references to fundamental instead
- Response to fundamental questions at the heart of the method
- No actual computational code [incomplete and empty, missing main part] or description of most computationally demanding details

I thank the authors for their answers. Regrettably, I still cannot recommend the acceptance of this method as a valid general real-space approach to Euler topology, on which I further elaborate below. As I am very well informed in this subject, I am a bit disappointed by some of the answers and especially the fact that some of my insights were twisted.

A case in point is the *whole* construction, i.e. the core of the paper (even at the analytical level), and a relation to the Pfaffian invariant. As I pointed out repeatedly, while the question is interesting, the question is hard in the sense that the Euler invariant is a multi-gap invariant, as originally defined in the momentum space formalism. There, on taking a patch $\mathcal{D}$ in the BZ the invariant reads $\chi = \frac{1}{2\pi} \int_{\mathcal{D} \in \text{BZ}} d ^{2} \textbf{k}~\text{Eu} - \frac{1}{2\pi} \oint_{\partial\mathcal{D}} d \textbf{k} \cdot \vec{a}$, where $\vec{a}$ is the Euler connection defined in terms of the Pfaffian of the non-Abelian Berry connection $\vec{A}_{n,n+1}(\mathbf{k}) = \langle u_n|\nabla_{\mathbf{k}} u_{n+1}\rangle $ and $\text{Eu} = \nabla_{\textbf{k}} \times \vec{a}$ is the Euler curvature [Nature Physics vol. 17, p. 1239–1246 (2021)]. So crucially the Euler form takes/combines/compares *different* states, by taking an inner product between two band states that form the two-band subspace. Now the crucial question is how is this multi-band/multi-state form defined in real space WITHOUT translational symmetry. Only when connecting to that limit the discussion makes sense, but this is not truly a marker as in the Chern case [\textit{Note that here this problem does not arise the same way. For the Chern invariant, one considers the projector-reformulated curvature for individual states on comparing with themselves (hence arises the trace structure), i.e the diagonal components of position/derivative operators between the same states with the same band indices are considered, as the Chern invariant involves projectors onto single/individual bands, unlike the multi-band Euler invariant}]. Indeed if one does not know the bands [requiring translational symmetry] how does one disentangle the state states in the real-space Pfaffian? For a true (general/universal) marker one is given the disordered system and should evaluate THIS system to see non-trivially, as is done for the Chern marker without the knowledge of the states in translationally invariant case.

It was precisely in view of the above point that I checked how this is done in the code. I am surprised that the authors asserted that I did not check whether "pf" is a "standard Python module" or not, before providing my criticism. The list of "standard Python modules" is available here: https://docs.python.org/3/py-modindex.html. Alas, the imported "pf" is not listed amongst the index of "standard Python modules", which makes the method questionable to me, to say the least. Then one notices that apart from a simple commutator the code calls a function "pf" that is not defined, one could try to "pip install pf", in case it was not a standard module, as the authors claimed. The installed module (https://pypi.org/project/pf/) turns out to be the "Project Files templating engine"... Now, "pf" can be called indeed. However, with the importation implemented by the authors, it calls not for a Pfaffian, but for the "project file" (standing for "pf") management.

Apart from the ironic fact that there seems to be no standard "pf" function in Python, my question revolved around how one defines the Euler invariant [separating the states if there is no translational symmetry]. While anyone can take a Pfaffian of an antisymmetric matrix (especially in momentum-space Bloch state basis, with ``$N_{occ} = 2$", as the authors mention), the whole point is that the authors claim they can compute a marker in absence of well-defined momentum states (I am not even convinced if in a real-space basis, the operator on which the Pfaffian is supposed to operate is generally antisymmetric in the real-space basis within the proposed approach), which is needed to compute the invariant in a concise manner, refering only to the real-space eigenstates. This is completely missing as a whole in the code, which apart from simple computing a (standard) commutator (as for the Chern markers...) gives no insights.

Moreover, it should be stressed that the computation of Pfaffians in and extended, such as real-space, basis (for a real-space Euler invariant, rather than the marker) is numerically a *very* costly task [ACM Trans. Math. Software 38, 30 (2012)]. Namely, rather than computing a "square root of a determinant" for a $2 \times 2$ matrix (in momentum-space); here, one needs to compute a "square root of a determinant" for, say, a $2000 \times 2000$ matrix, to get the Pfaffian. I am not quite sure how the unspecified "standard Python module" can handle that serious task.

Similar issues arise with their answer referring to the work of Bellissard. Again, I am familiar with this high end work, as well as the work of Connes. The point is that using the notions of non-commutative Brillouin zones generalizations of Chern numbers [again single bands] are introduced. It is proven that they are invariants that doesn't mean that without any further insight one can repeat this for the multi-gap Euler case. The whole point is that this has to be shown to be an adequate setup and producing a marker as claimed.

Also in view of the reference to the work of Bianco and Rsta I must note an important difference. First as there they consider the single Chern band case one can effectively compute the commutator and show these sum (with a trace, rather than a Pfaffian) to the Chern number. It is furthermore also well established in the context of Thouless pumping and the modern theory of polarization how these concepts hang together (based on the trace structure). Again the multi-gap nature in which one has a trace over a Pfaffian over two states, which involves mathematical operations that do not commute, unlike a combination of two traces, makes it even more complicated. The question of how to define this in absence of well defined bands as function of momentum is an aspect that is the core of the problem and not answered at all by the authors. They just claim this can be done, although not clear how, other than referring to very general statements like the ones above, incl. the use of "standard Python modules" to compute the troublesome Pfaffian.

Finally, I stress that the authors should note the role of multi-gap topology. Fragile topology means invariants beyond K-theory that can be trivialized by adding trivial bands, contrary to stable invariants such as Chern numbers or $Z_2$ QSHE invariants that need a gap closing with bands having opposite invariants. However fragile topologies can [and indeed the first paper by Po, Watanabe, and Vishwanath is an example] can be symmetry indicated. In that case they are simply the difference between two elementary band representations that represent an atomic limit. The Euler class is arising in a two-band subspace due to the partitioning of bands [Physical Review B 102, 115135 (2020)] and the value can be seen as being changed due to the brading of band nodes between different gaps [Nature Physics 16, 1137-1143 (2020) and Physical Review Letters 125, 053601 (2020)] this can be completely symmetry indicator-free-i.e for bands with all same irreps (as in the case of trivial bands). This is not really addressed by the presented real-space approach, and is not reflected in the paper.

All in all, given the other reports, I anticipate that this paper will be (and probably can be) published, but I think that these reservations should at least be mentioned and acknowledged, rather than being swept under the rug with references to general well established theories that do not a priori apply here. Indeed, I find it rather obscure of how the authors retrieve such impressive numerics with the non-existing "standard Python module" applied within a (put-mildly) questionable real-space approach, to such a complicated problem.

see report; main issue is to fix or at (*least acknowledge*) the problem with defining the Pfaffian marker in real space without referring to translationally invariant case.

Ask for minor revision

I thank the authors for carefully addressing my questions and suggestions. In the revised manuscript, the authors have added the discussion on composite Wannier functions and the corresponding ordering, and improved the presentation of the results. Since the results are very interesting and important, thus making a significant advancement in the field, I can recommend the manuscript for publication in SciPost Physics.

Publish (meets expectations and criteria for this Journal)