Amorphous topological phases protected by continuous rotation symmetry

In the manuscript, the authors show that a topological crystalline insulator can be protected by average reflection symmetry and average continuous rotation symmetry in an amorphous system. In stark contrast to a topological crystalline insulator in a crystalline system, the protection is promoted to every edge orientation. In addition, the authors give a classification of topological phases in continuum and amorphous systems protected by continuous rotation and unitary reflection symmetry. Finally, the authors briefly discuss the possible realization of amorphous systems with Shiba glasses, cold atoms, photonic crystals and metamaterials. The paper indicates that amorphous materials can provide a protection for gapless edge modes by average symmetries, which are absent in crystalline materials, thereby significantly advancing our understanding of the disorder effects on topological phases in amorphous materials. The paper is also very interesting to read and thus I can recommend it for publication in SciPost with some questions and comments.

1. In the introduction, the authors introduce a fact that higher order topological phases can also appear in amorphous systems. I would like to mention a paper arXiv:2012.12052 where second order topological phases have also been found in amorphous systems. In fact, there, structural disorder induced higher-order topology has also been predicted.

2. In the fourth paragraph of page 3, the authors state that “We expect to find protected gapless phases in the presence of strong disorder in symmetry classes …”. I wonder whether strong disorder respects some symmetry or average symmetry.

3. In the second paragraph of subsection A. Continuum models in Sec. V, the authors consider a class D Hamiltonian to explain the classification based on the reflection symmetry and particle-hole symmetry. I think that it is only when the Chern number is zero that one needs to classify the phase by considering the reflection symmetry. Can the authors clarify this point?

4. In Table II, the authors display the classification for continuum and amorphous systems. I think the authors should explain in more detail why the classification is different for amorphous and continuum systems. For example, in the class AIII, why does the classification become $\mathbb{Z}_2$ in amorphous systems compared with the $\mathbb{Z}$ classification in continuum systems.

1. relevant and timely topic
2. appropriate technical approach
3. systematic and general approach
4. significant new conclusions

1. the main message could have been summarized and communicated earlier and more efficiently, however, in general the presentation is good

The authors study the fate of reflection-protected crystalline topological states in 2d amorphous systems. The work employs methods of topological band theory together with numerical calculations in lattice models. The overall conclusion is that, in contrast to crystalline systems, amorphous structures exhibit robust edge modes for generic edge terminations. However, the transport properties are not quantized, rather they reflect those of a critical 1d system. The work is well-documented and no doubt technically correct. It illuminates the topological properties of random lattices which is timely and interesting topic. Therefore I recommend that the manuscript is published. However, I would like the authors to address some questions and consider if they would like to clarify some aspects of the work before publication.

1. How can the notion of amorphous structure and protection by average continuous rotation be quantified a bit more formally? Does average rotational symmetry imply that the spatial correlations of lattice sites should be rotationally invariant? Is this the operational definiton of amorphous structures employed in the manuscript?

2. Related to the previous question, one can imagine random graphs that have residual spatial correlations, for example regular lattices where lattice sites are perturbed from their original positions by random displacements. Intuitively, one would guess that when the random displacements are of the order of the lattice constant, this would effectively correspond to average rotational symmetry. Is this correct? Would residual correlations open small gaps?

3. As far as I understand, the main rationale of the work is the following. First, topological states protected by reflection are identified in translation-invariant continuum systems. Then, the representative Hamiltonians of these classes are discretized on random lattices. This leads to conclusion that for generic non-high symmetry terminations, random systems exhibit protected edges, in contrast to translation-invariant systems. However, the transport is no longer quantized but rather shows critical scaling. Personally, I'd like to read the summary of the main conclusion as early as possible (abstract&intro). This is matter of taste but I feel that findings are now buried into convoluted technical parts. In the perfect world, people would read each others papers and, thus, appreciate the results sprinkled in the text. However, we are not living in the perfect world.

4. This could be my personal confusion but I would like to have one conceptual issue clarified. In some sense, the Hamiltonian and the lattice structure both arise from interactions of the microscopic constituents. Luckily, we can often think them separately in the sense that we can regularize different model Hamiltonians on whatever lattice. So, while it is completely ok to study the reflection-protected models on random lattices, there does not seem to be a compelling reason why the symmetry-forbidden terms in crystalline lattice would locally vanish in amorphous systems. For example, if we compared a crystalline and amorphous phase of the same material, is it true that the amorphous phase could still exhibit the reflection-forbidden terms as disorder?

5. I suppose random systems are always gapless in principle. At high density (in the units of the effective hopping radius), the probability of exponentially suppressed density fluctuations are just so tiny that these rare-region effects do not show up in finite size numerics. But in principle, in the thermodynamic limit, there would exist Lifshitz tails due to empty patches of lattice points (if this is not explicitly forbidden by the model). Does this affect, in principle, the validity of the topological invariants introduced in the manuscript?

I have indicated point 1-5 above for the consideration of the authors. However, I would leave the appropriate actions to their own discretion.