Geometry and Topology Tango in Ordered and Amorphous Chiral Matter

Detailed answers to your questions and comments:

1) The pencil matrix method to estimate the Wannier centers without performing the costly optimization of the spread functional is of great interest to the community. However, whether this method is generally applicable is a highly nontrivial question, that the manuscript does not address, beyond comparing the results in a few specific cases and finding good agreement. Besides a strong argument for the method's applicability missing (with reference 51 to an entire book being too vague), it is also unclear what exactly the method is. I thank the authors for making their Mathematica code available, however, this code only calculates results for a single choice of α's, while the results in the manuscript are obtained by averaging over an ensemble. What is the distribution the α's are drawn from? How are the position eigenstates for various α values matched while averaging? Considering that one choice of α's corresponds to calculating the projected position eigenvalues along one randomly chosen axis, it is not obvious which eigenstate corresponds to the same Wannier center in a large sample for different choices of α's.

We do agree with you. We did not provide enough details about this original method in our first version of the main text. We thank you for suggesting a more detailed presentation of this new method. Following your advice, we have included a detailed discussion of the matrix pencil method in Appendix D and added a new Figure (Fig. 11). This discussion shows how to suitably choose the parameter $\alpha$ in periodic lattices and how the computational cost of this method compares with the conventional computation of Maximally Localized Wannier States.

The Mathematica file we upload was indeed incomplete. We have corrected this unfortunate mistake. Thank you for having pointed it out.

2) The manuscript considers systems with a chiral symmetry of the sublattice type, i.e. not every site contains an equal number of even and odd orbitals under chiral symmetry, but the chiral symmetry arises from the bipartite nature of the lattice. I am not convinced of the physical relevance of this case for amorphous matter as studied in sec 5.3, as naturally occurring amorphous lattices are rarely strictly bipartite.

We agree with you in the context of solid state physics, where chiral symmetry typically emerges at low energy. However, we stress that photonic metamaterial can be chiral by design and that all mechanical structures enjoy an intrinsic chiral symmetry as first revealed by Kane and Lubensky (Nature Physics 2014). Remarkably this features holds both in homogeneous and heterogeneous mechanical metamaterials. Each positional degree of freedom (A site) is only connected to a stress degrees of freedom (B site). We stress on this crucial point in the main text (Introduction of Section 5 line 285-291)

1. There seem to be several typos related to prefactors included in the definition of various quantities, and the gauge ambiguities of them. For example in (39) m∈πZ is clearly a typo and should be m∈Z. Moreover, some definitions seem to be inconsistent, for example using the definition (4) of γ^A/B, going from (37) to (38) a factor of 2 seems missing. Overall it is quite confusing that the definitions in Appendix A.3, B.1 and B.2 contain extra factors of BZ volume compared to those used in the main text. It is often hard to decide whether the k integrals are over the volume of the BZ, or along 1D lines. I ask the authors to unify the notation and carefully check for typos.

Thank you for you thorough reading of our manuscript. We have corrected all the typos you found and have carefully double checked all of our notations:

“For example in (39) m∈πZ is clearly a typo and should be m∈Z.”: 
We corrected it.

“Moreover, some definitions seem to be inconsistent, for example using the definition (4) of γ^A/B, going from (37) to (38) a factor of 2 seems missing.”:
There is no inconsistency in this definition. (4) corresponds to the sublattice Zak phase of one energy band (indexed by n). However, from (37) to (38) we use the sum over all the bands, giving twice the total sub lattice Zak phase.
 “Overall it is quite confusing that the definitions in Appendix A.3, B.1 and B.2 contain extra factors of BZ volume compared to those used in the main text. It is often hard to decide whether the k integrals are over the volume of the BZ, or along 1D lines. I ask the authors to unify the notation and carefully check for typos.”: 
We did check for typos and misprints to make our manuscript and Appendix as clear as possible. To avoid any possible confusion we now use a consistent notation for the integrals over the BZ and we chose this notation as distinct as possible from the contour integrals defining the Zak phases.

1. There are several references to "Methods", these should be replaced with references to specific Appendices.

We have done so.

1. The term "atomic limit" is used throughout the manuscript without defining it. While it is clear from the context for experts on the topic, this terminology may be confusing, as the atomic limits in question are in fact perfectly dimerized limits. I ask the authors to clarify the terminology in the manuscript. It is also worth noting, that in more complicated systems, with more types of (further neighbour) hoppings, it may be less obvious what the atomic limit of a given system is, it would be useful to give a more general definition.

Following your suggestion, we have added a formal and a practical definition of the atomic limit in Section 3:

« We recall that the atomic limit of a material corresponds to a smooth deformation of the couplings to separate the energy scales so that so that the Wannier functions are exponentially localized, and respect the symmetries of the crystal~\cite{Bradlyn2017}. in practice, it consists in choosing a unit cell including the strongest couplings. » (We also refer to the work of Bradley et al.)

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