Peer review of: Real-space recipes for general topological crystalline states (2020)

Reviewers' Comments: 
Reviewer #1: 
Remarks to the Author: 

Topological phases protected by spatial symmetry is recently one of the most active topics in 
condensed matter theory. This article provides a general recipe to classify topological symmetry 
with spatial and internal symmetry together for both fermionic and bosonic systems. In my 
opinion, it is an important development in the field and therefore deserves publication in nature 
communication. That being said, I have to also say that the article is quite technical in nature. 
Therefore, I have a few minor comments on the presentation of the paper. 
 
1. In Fig. 2(b), the example given here, does it assume periodic boundary condition? 
 
2. In the HOLSM section, the authors first give some pictures about LSM of translational symmetry 
and projective rep. Later (on page 15), it is generalized to other symmetries. Can the authors give 
an example for a more general case beyond just translation? 
 
3. My impression after reading is that the general strategy for classifying boson and fermion TCS 
are the same. The only difference is just that one needs to decorate boson/fermion SPT on the p-
blocks for bosonic/fermionic TCS respectively? Is this statement correct? Can the authors stress 
more about which part of the classification scheme is general for both boson and fermion systems 
and what is the essential difference between them? 
 
4. On page 13, the authors discuss an example that is beyond layer constructions. The authors 
state that “Therefore the classification of bosonic TCS in this case is identical with classification of 
free fermion topological crystalline insulator. ” I cannot quite make sense of this statement. It 
would be great if the authors can explain a little on this. It would also be clearer if the authors 
provide this example beyond layer construction explicitly. 
 
Overall, I think this paper has many interesting and original results and should be published in 
nature communication after minor adjustments. 
 
 
 
Reviewer #2: 
Remarks to the Author: 
The authors claim that they present a unified scheme for constructing all topological crystalline 
states, bosonic and fermionic, free and interacting. The key idea is to uese the so-called real-space 
“building blocks” and “connectors”. Building blocks are finite-size pieces of lower dimensional 
topological states protected by onsite symmetries alone, and connectors are “glue” that complete 
the open edges shared by two or multiple pieces of building blocks. Then they applied this scheme 
to obtaining the full classification of bosonic topological crystalline states protected by several 
onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space 
groups in three dimensions. They further claim that their real-space construction gives the 
complete set of topological crystalline states for bosons and fermions. I found that the main results 
of this manuscript are very interesting, and it warrant a publication in some form. However, I also 
feel that the presentations are two mathematical, and some claims are too "big". 
 
(a) Does the arrow in Eq. (2) means a exact sequence? If it is indeed the case, please carefully 
explain the notation and explain why such a mathematical notation is necessary for understanding 
this physical problem. 
 
(b) The terminology "page" below Eq.(2) is borrowed from the mathematical concept of spectrum 
sequence. Such a word could be very strange for readers who are not aware of spectrum sequence 
theory. It is better to start from a physical example or physical picture for the construction instead 
of heavy mathematical notation. Necessary mathematical background could be put into appendix 
or supplementary material. I believe many readers could get lost once they hear about "pages". 
 
(c) Although the results are well tested for interacting bosonic systems as well as some free 
fermion systems, there is no single example of interacting fermion system. In particular, I believe 
that the "obstruction" and "trivialization" conditions are much more complicated for interacting 
systems, provided that the classification scheme for on-site symmetry are already very 
complicated in interacting fermion systems. Therefore,I believe that the statements such as "for 
constructing all topological crystalline states, bosonic and fermionic, free and interacting" and 
"real-space construction gives the complete set of topological crystalline states for bosons and 
fermions" might be over claimed. 
 
 
 
Reviewer #3: 
Remarks to the Author: 
Recently, there have been quick developments in the understanding of crystalline symmetry 
protected topological (SPT) phases. One of the systematic approach, which is usually called “block 
state” or “topological crystal” picture, conjectures that crystalline SPT phases are adiabatically 
connected to stacks of lower-dimensional SPT phases arranged in some pattern respecting 
crystalline symmetry. These lower-dimensional SPT phases should satisfy a ``gluing condition” 
such that all the gapless modes in the bulk should be glued together and gapped out while 
preserving symmetry. Since there are many block states which can be deformed into each other, 
one then needs to consider all possible deformations as equivalence relations in order to obtain the 
classifications. 
 
The original proposal is based on physical arguments, and the formal mathematical structure was 
not clear at that time. There are three papers that appeared around the same time which revealed 
that the relevant mathematical structure should be the spectral sequence in some generalized 
homology/cohomology, and this paper is one of them (the other two papers are Phys. Rev. B 99, 
115116 (2019), arXiv: 1810.00801). Compare to the other two papers, this paper focuses on 
phases within group cohomology and works through the details of the spectral sequence explicitly. 
Moreover, the authors show that many of the computation in the spectral sequence can be made 
algorithmic. For example, the authors show that there is an explicit formula of ``gluing” for 
phases within group cohomology. Such computations had to be done by considering physical 
models and see how to glue these models by adding symmetry allowed couplings. With this 
formula of ``gluing”, the calculation is significantly simplified and can be made automated, which 
is done in this paper for bosonic crystalline SPT phases with several internal symmetry groups and 
all the wallpaper groups and all the space groups. The authors also discuss an application of this 
approach: deriving generalized Lieb-Schultz-Mattis constraints, which stating that, in some 
situations, a symmetric, gapped, unique ground state must be a non-trivial SPT state. This paper 
is also well-written and easy to follow. Readers who are less comfortable with formal mathematics 
might find this paper more accessible. I do not have particular concern for this paper. I 
recommend this paper to be published in Nature Communications.