Average Symmetry Protected Higher-order Topological Amorphous Insulators

1) Clearly written, good presentation.
2) Discusses the timely problem of amorphous topological phases, and more broadly, topological phases protected by non-crystalline spatial symmetries.
3) Novel use of "momentum-space" invariants derived from twisted periodic boundary conditions with more k components than spatial dimensions.

1) My main concern is the novelty of the results compared to the earlier works on quasicrystals (refs 28 and 29), and the relevance of the results to physical amorphous materials. The p-fold rotational symmetries (particularly p=8, 10, 12) arise naturally in quasicrystals, but seem artificial in an amorphous system. Amorphous materials are typically isotropic, possessing continuous rotation symmetry on average, lacking any preferred directions. In my view, the systems studied here are better described as highly disordered quasicrystals - or perhaps an amorphous layer on a quasicrystal substrate - inheriting the discrete rotational symmetry in the hopping Hamiltonian, but the atomic positions completely disordered. Do the authors agree with this view? Can the authors suggest other physical setups that these models can describe? I believe that these consideration should be discussed at some point of the manuscript to make the context of the research clear to the reader, and motivate the relevance of the results to physically realizable amorphous materials.

2) What guarantees that there are no gapless modes at the boundary of the segments after restoring the rotational symmetry when constructing the Hamiltonians H_j, or when applying periodic boundary conditions? As described in the text, the amorphous structure is not completely uncorrelated, rather it is a random set of hard disks. Is this constraint on the minimal interatomic distance obeyed by the symmetrized systems? Based on the description of the construction, it is not, which may result in additional sub-gap states at the gluing interfaces in the symmetrized and periodic boundary condition systems. I ask the authors to clarify these questions and demonstrate that no new subgap physics arises at the gluing interfaces, for example by examining the spatial localization of the low-energy states that appear at large system sizes (fig 3c inset).

3) Why does the invariant chi_n not depend on n? The manuscript only presents numerical evidence for chi_1 = chi_3. I would expect a similar scenario as in ref 28 SM, when additional symmetries forcing a vanishing Chern-number are responsible for this equality, and is not forced by protecting symmetries. It would be worth to check whether this is the case, or at least comment on the possibility.

4) Why is it necessary (or advantageous) to distort the system to calculate the quadrupole moment? Is the quantity measured this way really the quadrupole moment, or perhaps some higher moment? It is really unclear to me how a procedure like this, manifestly breaking the protecting rotational symmetries, can extract a topological invariant.

The manuscript studies topological phases of amorphous systems protected by chiral and average p-fold (p even) rotation symmetries. The analysis is carried out using a tight-binding model on a random graph, using "momentum-space" invariants derived from twisted periodic boundary conditions, quadrupole moment, and spectral signatures. The manuscript is well written, and the results are sound.

In my opinion the manuscript is an interesting, but fairly straightforward generalisation of earlier work on quasicrystalline topological phases with added disorder (see point 1 of the Weaknesses section for details). Hence I recommend moving the manuscript to SciPost Physics Core, and publication with minor clarifications; or would request the authors to further support the novelty of the work, and motivate why the paper meets the acceptance criteria of SciPost Physics in light of the questions raised.

Minor change requests below, for major questions see the Weaknesses section.

1) The hopping Hamiltonian in eqn. 1 for p=8 is (up to a basis transformation, and inclusion of bond-length-dependent prefactors) is identical to that of ref [28], and the generalisation of the last term for cases with p other than 8 was presented in ref [29], I ask the authors to make this clear by citations when introducing the Hamiltonian.

2) The following statement is unclear to me: “Note that near mz ≈ 0.3 there appears an intermediate region with the coexistence of topologically nontrivial and trivial samples where the gapless bulk states are extended as shown in Fig. 3(d).” Should clarify the text or fig. 3 to make this clear.

3) Should mention in the caption what the vertical dashed grey lines in fig. 3 denote.

4) "high-symmetric momenta" should be, following standard terminology, "high-symmetry momenta"

The paper shows that higher order phases which are protected by rotational symmetries - which cannot be achieved in crystals - and were shown to be realised in quasi-crystals can also be realised in amorphous systems.

1) To that effect the authors show that while the boundary modes get realised in any typical system; to well-define a topological invariant they need to make some equivalent systems where the lattice is deliberately made rotationally invariant and a band structure can be made by artificially repeating the pattern. They show that the region where the Pfaffian of this system is non trivial - is consistent with non-trivial phase of the system and associated gap closings.
2) They also calculate the real space quadrupolar moment in the system by deforming the lattice into a square patch which is consistent with these transition values.
3) To further investigate the low energy modes near half-filling they look at typical level spacing ratio and find that transitions between topological and trivial phases are characterised by change in ratio from GOE ensemble to poissonian which is interesting.

No particular weaknesses

I find the paper interesting and recommend publication. I particularly find the momentum space construction and the invariant calculation interesting and it might be useful for the community of researchers working on topological phases away from crystalline settings. I was wondering if the IPR of the states close to Fermi energy show any anomalous scaling near the transitions as a function of $m_z$. It will be useful if the authors can comment on that.