A $C^\ast$-algebraic view on the interaction of real- and reciprocal space topology in skyrmion crystals

Dear Referee,

We want to thank you for your objective and kind assessment of our work and for the helpful comments, which speak of the time and effort dedicated to the review of our manuscript. Below, we address how the manuscript has been modified as a response to the issues raised.

1. I felt that Teo and Kane https://arxiv.org/abs/1006.0690 deserve some credit for the idea that spectral properties of noninteracting fermions are sensitive to both the topology of the torus associated to twisting boundary conditions and the topology of defective background fields.

This reference describes a framework for the classification of topological defects of varying dimensionality and their impact on the emergence of protected gapless modes through the lens of the bulk-boundary correspondence. This is closely related to sections 5.2 and 5.3 of our manuscript, in which we observe band crossings while passing a point defect in a three-dimensional space, comprising two spatial dimensions and one parameter dimension, as outlined in section 2.3. We feel it is only appropriate to include this reference and acknowledge the authors' work on this kind of correspondence.

1. Do I understand correctly that $r$ in Eq. (5) is defined by the range of $\bf{m}\in\mathbb{Z}^d$ such that the Fourier coefficient $\bf{n}_{\bf{G}_{\bf{m}}}$ is either zero or smaller than some threshold. As such, it can be larger than the dimensionality $d$ of space. Perhaps this could be emphasized around Eq. (5).

It is correct, that $r$ may exceed the dimensionality $d$. However, the truncated texture in Eq. (5) is supposed to consist of the lowest order Fourier modes without a restriction on the norm of $\bf{n}_{\bf{G}_{\bf{m}}}$. In the regime of multi-$q$ textures it is expected that these modes dominate the higher order terms. In the current form of Eq. (5) $\bf{m}$ may take values in ${-1,0,1}^d$ and thus it is $r\leq 3^d$. To be consistent with the $r$ occurring in the trigonometric formulation in Eq. (6), we modified Eq. (5) to

$\bf{n}(\bf{x})\approx \sum_{i=-r}^r \bf{n}_{\bf{q}_i} e^{i(\bf{q}_i\cdot\bf{x}+\varphi_{i})}$

with $\bf{q}_{-i}= -\bf{q}_i$, $\bf{n}_{\bf{q}_{-i}}=\bf{n}_{\bf{q}_{i}}^\ast$, and zero mode with $\bf{q}_0=0$, $\varphi_0=0$, and $\bf{n}_{\bf{q}_0}\in\mathbb{R}$. In this form, $r$ is may take values up to $(3^d-1)/2$.

1. I believe that $\bf{l}$ belongs to $\mathbb{Z}^d$ not $\mathbb{Z}$ after Eq. (16).

Indeed, it was supposed to read $\mathbb{Z}^d$.

4.Does not the introduction of the $r$-torus presume that $\bf{q}_i$ with $i=1,\dots,r$ are all linearly independent, which, however, is not necessarily the case? Would it not be cleaner to introduce $\tilde{r}$ before Eq. (18) instead of after Eq. (18)?

In Eq. (18) the $r$-torus is introduced as the ambient manifold of the orbit. Therefore, the linear independence of the $\bf{q}_i$ does not need to be presumed. We agree, however, that it is a suggestive notation that should be amended. We opted to add a paragraph at the end of section 2.1. to introduce $\tilde{r}$ and to address the differences between periodic and aperiodic textures. This way we only need to discern two cases for Eq. (18).

5.Should it not be $\bf{x}$ as the argument of $\hat{\bf{n}}$ instead of $\boldsymbol{\omega}(\bf{x})$ in the first equality of Eq. (26)? Perhaps it could be said that the second quality of Eq. (26) follows from Eq. (17).

With a misuse of notation, we use the same symbol $\hat{\bf{n}}$ for the texture defined on real space and on the phase torus: $\hat{\bf{n}}=\hat{\bf{n}}\circ\boldsymbol{\omega}$. We added an explanation for this identification, which is the continuum version of Eq. (17), to the beginning of section 2.2., and refer to it for Eq. (26).

6.I would have traded (33) for $\bf{m}\mapsto\alpha_{\bf{m}}$ whereby old Eq. (33) becomes Eq. (34).

We added $\bf{m}\mapsto\alpha_{\bf{m}}$ to Eq. (32).

7.Should one not set $[t_i,t_j]=0$ for $i,j=1,\dots,d$ in Eq. (34) in order to get the group $\mathbb{Z}^d$?

The relator was meant to indicate that $[t_i,t_j]$ is equal to the group identity in this presentation. For improved clarity, we changed it to the relation $t_i t_j = t_j t_i$.

8.Before Eq. (37), reference is first made to a Banach Algebra and then to a complex Banach Algebra. I am guessing that the adjective complex refers to the underlying field. Perhaps, "over a field" can be added to the first instance of a Banach Algebra.

This is correct. We added "algebra over a field" at the first instance.

9.Should one not set $[u_k,u_l]=0$ for $k,l=1,\dots,r$ in Eq. (47) in order to endow $C(T^r)$ with a Abelian group structure under point-wise multiplication?

Same as for comment 7), we changed it to $u_i u_j = u_j u_i$.

10.Why is $p$ chosen to be a prime number in Eq. (102)? Is it because the normal subgroups should be fields as well?

$p$ only needs to be a natural number greater than one for this construction to work. The choice of $p$ being a prime number is part of the example and carries no significance. We changed it to the more general assumption.

11.I did not understand the sentence "There are several clear gaps, which are connected to the adiabatic limit ...''

With the adiabatic limit, we refer to systems with $\vartheta\to0$ for which the slow spatial variation allows for the adiabatic assumption that the electronic states remain in the same energy state. In general, the adiabatic criterium also depends on the coupling parameter $\lambda_\textrm{xc}$, the lifetime of the itinerant electrons, and the Fermi velocity. However, in the strong coupling limit, we can assume that adiabaticity is realized for large skyrmion size.

In the density of states figure Fig. 3(a) there are several gaps, some of which are only open over a short interval of finite $\vartheta$. With gaps connected to the adiabatic limit, we refer to the once, which shrink in width for $\vartheta\to0$, but only seem to close at $\vartheta=0$. Between some bands of higher density the would-be gap is filled by states with low density. Therefore, with "clear" gaps we meant emphasize that the gaps are free of any states.

For the resubmission of the manuscript rewrote the aforementioned sentence to make clearer to which features of the figure we refer.

12.Figure 3 contains a Figure in panel (a) and two tables with panels (b) and (c). It was a little confusing to call Tables in the text. Perhaps Panel (a) could be made a stand alone Figure, while panels (b) and (c) could be made stand alone Tables.

We agree, separating the figure from the tables improves the layout.

Sincerely,

Pascal Prass and co-authors