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

We were happy to read the Referee's positive report and thank the Referee for reviewing our manuscript. We address the two comments below:

Comment 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.”

Response 1: We thank the referee for this suggestion. Accordingly, we have added a remark in our MS:
“Formally, Eq. (6) share a similar expression to the real-space Chern number except for the substituting from Tr to Pfocc.”

Comment 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]?”

Response 2: We thank the referee for this important question.

(1). Translation to Real-Space Expressions:
The reality condition of u(k) is crucial for the Euler number, as discussed in previous literatures (For example, Section C of the supplementary material of [Nat. Phys. 16, 1137–1143]). For any spinless time-reversal invariant model, the time-reversal operator T satisfies: (i) $T^2=1$ and (ii) $THT^{-1}=H$.
After the Takagi decomposition, we can always find a basis in which the Hamiltonian is a real symmetric matrix. A real symmetric matrix is orthogonally diagonalizable, meaning its eigenfunctions can also be taken to be real.

(2). Required Symmetries and Implications:
The antisymmetry of the operator $P[UPU^\dagger, VPV^\dagger]$ is ensured by the reality condition and the expression itself. Specifically, for the submatrix A: $A_{mn}$=$A_{mn}^*$=$A^\dagger_{nm}=-A_{nm}$. This implies $A=-A^T$, maintaining antisymmetry even in the absence of spatial translational symmetry.
To summarize, the reality condition ensures that the eigenfunctions $u(k)$ can be taken as real, leading to a skew-symmetric field strength matrix. This antisymmetry is preserved in the real-space formulation through the structure of the operator $P[UPU^\dagger, VPV^\dagger]$, thus allowing the definition of the Pfaffian and ensuring the topological nature of the Euler number in real space.

We would like to thank the referee for his/her high assessment of our work. We respond to the comments below and make corresponding changes in the revised manuscript.

Comment 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.”

Response 1: We thank the referee for reading our manuscript carefully. Accordingly, we have made the following revisions in our MS:
“where |r> denotes the basis to construct the external space indexed by the Wannier cell r.”


Comment 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.”

Response 2: We thank the referee for this important question. The anti-symmetry required for the definition of Pfaffian is indeed ensured by the structure of the expression itself. To clarify, we present the submatrix (denoted as A here in both k and r space) in its original form as shown in Eq. (2) in the main text, which can be written as $\langle\partial_{[k_x}u_1(k)|\partial_{k_y]}u_2(k)\rangle$. Due to the commutator, an additional sign appears from the interchange of 1 and 2, ensuring the anti-symmetry of the submatrix. Therefore, considering $A_{12}$, one off-diagonal element, effectively defines the Pfaffian.

This discussion extends to Eq. (6), where the submatrix $A=\hat{P}[\hat{U}\hat{P}\hat{U}^\dagger,\hat{V}\hat{P}\hat{V}^\dagger]$ in r space also involves a commutator. Specifically, $A_{mn}=A^*_{mn}=A^\dagger_{nm}=-A_{nm}$, implying $A=-A^T$. Therefore, the submatrix inherently preserves anti-symmetry even in the absence of spatial translational symmetry.


Comment 3: “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.”

Response 3: We thank the referee for this important question. Analogous to the k-space case, the order of the Wannier basis can be generally determined by the diagonal elements of the Hamiltonian in this basis. Additionally, any sign change will manifest in the distribution of the local Euler markers. Hence, the order can also be determined by ensuring the continuity of the local Euler markers, thereby maintaining consistency throughout the calculations.
Accordingly, we have added a paragraph at the last of Appendix H.5 in our MS:
“Another issue to be clarified is the ordering of occupied states within a certain cell r. It can be determined by the corresponding diagonal element of the Hamiltonian on the composite Wannier basis. We also noticed that the local Euler marker is attached to only an additional minus sign when this ordering is inverted. Therefore, more conveniently, the sign of local Euler markers can be set to satisfy the continuity of these markers.”


Comment 4: “I think Fig. S3 and Fig. 1(d) should use logarithmic coordinates to better illustrate the power-law decay of (1-|e|) versus L.”

Response 4: We thank the referee for this helpful suggestion. Accordingly, we have updated Fig. S3 and Fig. 1(d) in our MS to use logarithmic coordinates.


Comment 5: “Please provide the specific relation between $t_{\mu\nu}$ and ($V_{dd\sigma}$, $V_{pp\pi}$, ……).”

Response 5: We thank the referee for this suggestion. Accordingly, we have added all these specific relations in Appendix H.1 in our MS.


Comment 6: “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.”

Response 6: We thank the referee for this suggestion. Accordingly, we have added a new section with a new Figure S8 in Appendix I.7 of the MS:
“In this section, we discuss the upward shift of eigenenergies of corner states with increasing on-site energy. As illustrated in Fig. 3 and Fig. S7, introducing structural disorder leads to the upward shift of the eigenenergies of corner states. In fact, this effect originates from the decreasing energy gap. In structurally disordered samples, the decrease is attributed to the increasing disorder amplitude. Additionally, the adjustment of the on-site energy can also lead to a smaller bulk gap. As discussed in Appendix I.1, in region I, we can lift the on-site energy of p-orbitals, causing the bulk gap to decrease and eventually vanish at critical point $\Delta_1$. Therefore, for comparison purposes, we consider a square model with the parameters the same as in Fig. 3 except for the on-site energy difference. As illustrated in Fig. S8(a), increasing the on-site energy difference shows a similar upward shift effect to that observed in structurally disordered lattices. These shifted states near the upper bound of the PBC gap are spatially localized at four corners, as shown in Fig. S8(b). These results show the similarity between the effect of on-site energy difference and structural disorder on the upward shifting, which can be explained as the effect of the decreasing bulk gap.”


Comment 7: “Please correct the citation “appendix ??” in line 188.”

Response 7: We thank the referee for carefully reading our manuscript and pointing out this mistake. Accordingly, we have corrected this citation and added a new section with Figure S5 in Appendix I.4 of the MS:
“In this section, we discuss the deviation of r-space Euler number with OBC. The OBC case shows a similar linear dependence between 1/L and the numerical deviation $\Delta e=1-|e|$ with slower convergent behavior. This means that the OBC includes an additional effect which is up to order $O(1/L)$ as well. Notice that the Euler number is obtained by averaging the local Euler markers at all sites. Since the sites far from boundaries are supposed to preserve similar properties to those in periodic systems, such deviation originates from the sites close to the boundary, which contributes $O(L_{edge}/A)=O((L^2-(L-2)^2)/L^2)=O(4L/L^2)=O(4/L)$ as expected. Here $L_{edge}$ and $A$ are the number of sites in the boundary and the whole sample, respectively. This additional factor accounts for the slope approximated to 1/4 in Fig. S5, confirming the effect on r-space Euler number from the edge.”


Comment 8 : “$a_E/1−|e|$ should be written as $a_E/(q−|e|)$ in line 608 and Fig. S2.”

Response 8: We thank the referee for this suggestion. Accordingly, we have made the necessary revisions in the MS.


Comment 9: “The disorder parameter of the amorphous lattice in Fig. 3 is missing. Please add the information. 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.”

Response 9: We thank the referee for carefully reading our manuscript and pointing out these issues. The methods to introduce structural disorder in cases in Fig. 3 and original Fig. S6 (Now Fig. S7 in the revised manuscript) are different. In Fig. 2, we study a finite amorphous lattice constructed by assigning random site displacements away from their equilibrium position in an initial square lattice. Consequently, all spatial symmetries are broken. However, in Fig. S7, we specifically preserve the inversion symmetry, for comparison with the case breaking all spatial symmetries presented in Fig. 3 of the main text. As a result, we find that the real-space method works for amorphous lattices without and with inversion symmetry.
Accordingly, we have added the following information to the caption of Fig.3:
“In the amorphous lattice, each atom is assigned with a random displacement following the Gaussian distribution with standard deviation $\sigma = 0.2$.”
We also made the following revisions in Appendix H.3 of the MS to accurately describe the methods used:
“The amplitude d of atomic displacements are determined by Gaussian distributions with standard deviation $\sigma = 0.2$.”

We thank the Referee very much for his/her insightful reading of the paper and extremely constructive report. We have made some revisions and added some references following his/her suggestions. We believe it has improved the quality and clarity of the paper. Below we answer point by point and describe the changes made.

Comment:
“-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. ”

Response:
We appreciate the referee's insightful comments. It is important to note that the Chern marker, similar to the Euler marker we propose, is essentially a definition rather than a derivation. In Bianco and Resta's original paper [Phys. Rev. B 84, 241106(R)], the pivotal steps leading to the local marker are found in Eq. (3) and Eq. (9), which facilitate the transition from k-space to real space and from a global quantity to a local quantity, respectively. They explicitly stated that these equations “should be interpreted as definitions” of the corresponding quantities in the main text just below these equations. While their paper does not provide a rigorous mathematical foundation for the concept of a local Chern marker, the numerical results presented were compelling and have had a significant impact on the field. In this context, we believe that our manuscript similarly introduces a valuable new local marker, which provides a possible local description of topological order due to the “nearsightedness” of the ground-state density matrix, just as emphasized by Bianco and Resta in their concluding remarks.


Comment:
“-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.”

Response:
We appreciate the referee's detailed feedback. The theory of characteristic classes of vector bundles is indeed foundational for topology defined in k-space. However, this k-space formalism becomes inadequate when dealing with real space, especially in the presence of certain types of disorder. In such cases, non-commutative geometry, as introduced by J. Bellissard et al. [J. Math. Phys. 35, 5373–5451], serves as a robust mathematical framework. In their work, they defined the non-commutative Brillouin zone (BZ) to include the effects of disorder, thereby allowing topological quantities in k-space to be generalized to real-space equivalents.

Non-commutative geometry is thus well-suited for real-space topological invariants, such as the Chern number and Bott index, as demonstrated in the literature [Lett. Math. Phys. 112, 126]. It admits that the real-space expression may not provide an exact quantized value, however, the real-space formula is well-defined on a finite torus and coincides up to corrections of order $O(L^{-1})$ with the integer. Our definition of the local Euler marker is inspired by the discussions in sections III.A and III.B of the pioneering work of Bellissard, van Elst, and Schulz-Baldes [J. Math. Phys. 35, 5373–5451]. While our manuscript may not attain the same level of mathematical rigor as the referenced work [J. Math. Phys. 35, 5373–5451], we are confident it offers valuable insights and perspectives that will be of interest to both physicists and mathematicians interested in this field.

Accordingly, we have revised our manuscript to include the following statement:
“To generalize a formula of topological system defined in k space to its real-space form applicable to disordered system, a standard mathematical framework is the non-commutative geometry, which provides the duality (see the equivalence at least for translational invariant systems in Appendix C)”


Comment:
“-Very precisely, starting from the real-space Hamiltonian of a generic inhomogeneous system that satisfies the reality condition (e.g. preserving a C22T 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 Pfocc 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 Pfocc without a reference to the clean bands \ketu1k, \ketu2k, appears to be swept under the rug in the code provided by the authors (i.e checking the gitthub).
-Namely, where the subtle Pfaffian Pfocc 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.”

Response:
We appreciate the referee's detailed comments.

First, we want to clarify that the function "pf" in our code is simply a standard Python function used to numerically calculate the Pfaffian. It can be understood as extracting the off-diagonal element when $N_{occ}=2$, and it is unrelated to the construction of Wannier functions.

Given the potential confusion surrounding the use of Wannier functions, we will clarify the conceptual framework of our manuscript. Our work proposes a new local Euler marker, the average of which can be interpreted as the real-space Euler number according to the standard non-commutative geometry paradigm. However, this marker can only be calculated on the Wannier basis, necessitating the construction of such a basis in systems lacking translational invariance.

A straightforward approach to constructing real-space Wannier functions is through functional optimization, where the initial choice of the Wannier function is critical. In our case, this initial value can, in principle, be set arbitrarily. Nevertheless, for improved efficiency and convergence, the initial function is often chosen as the Wannier function derived from a translationally invariant system. Specifically, for disordered lattices, the initial Wannier function constructed through the Fourier transformation of Bloch functions from a perfect lattice serves well. This approach, however, becomes challenging for quasicrystals and amorphous systems. In our work, we derived the initial Wannier function for quasicrystals from a 16×73 rectangular lattice (as depicted in Fig. 2).

For fully non-periodic systems where gauge optimization is ineffective, other methods, such as the Iterated Projected Position (IPP) algorithm, should be considered as they do not require initialization [Phys. Rev. B 103, 075125].

Accordingly, we have made the following revisions to our manuscript:
“Since the real-space Euler number obtained in Eq. (6) can only be calculated on Wannier basis, a crucial step in the numerical calculation is to construct the Wannier function in systems even without the spatial translational symmetry.
One possible way to construct the real-space Wannier function is the functional optimization method.
In the functional optimization process, a key factor is the selection of the initialization. In our case, this is the initial value of Wn. In principle, the initial Wannier function can be set arbitrarily. However, to obtain a more efficient and convergent result, the initial function can be chosen as the Wannier function obtained in a translational invariant system. To be more specific, for disordered lattices, the Wannier function constructed through the Fourier transformation of the Block functions obtained in perfect lattice is a great initial function. However, it might be hard to find such a k-space analog in quasicrystal and even totally amorphous systems. In our work, the initial Wannier function of the quasicrystal is obtained from that of a 16×73 rectangle lattice in Fig. 2.
As for the fully non-periodic systems where the gauge optimization fails to work, other methods such as the Iterated projected position (IPP) algorithm are supposed to be considered without the initialization requirement [104].”


Comment:
“-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.”

Response:
We appreciate the referee's suggestion. We have now cited this paper in our manuscript to acknowledge the contributions of relevant prior research.


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

Response:
We thank the referee for this suggestion. In the manuscript, we have provided a real-space approach for the Euler class and applied it to explore the Euler topology in quasicrystal and amorphous lattices. According to the referee’s suggestion, we have replaced the formalism with an approach in the title.