Quasiperiodic Quadrupole Insulators

The authors report an interesting numerical observation of a reentrant topological phase in a BBH model with a quasi-periodic modulation of the hoppings, and support this observation through extensive numerical simulations.

- Relatively limited discussion and interpretation.
- Numerical evidence sometimes indirect.

Disclaimer: I am providing this report while also acting as the editor in charge of the manuscript.

Quasi-periodic Hamiltonians are at a boundary between ordered and disordered, and they combine features of both. For example, they can have both polynomially and exponentially localized wave functions, as well as approximately metallic bands.

The author consider the impact of quasi-periodicity on the topological properties of the BBH model. There are multiple ways to interpret the properties of this model:

- One can consider it a model with corner charges and no chiral symmetry
- One can consider it a model with chiral symmetry and sublattice-polarized zero modes

In both cases, the literature distinguishes the model as being protected by a $C_4$ symmetry, or two mirror symmetries. The model with chiral symmetry and $C_4$ is a higher order topological phase.

Quasi-periodic modulation of the hoppings, as done in the manuscript, preserves chiral symmetry, and breaks all spatial symmetries, except on average. There is a significant body of literature demonstrating that average symmetries of disordered ensembles are in many ways similar to the exact ones, and in particular a recent Ref. https://arxiv.org/abs/2412.01883 provides a broad framework for those phases.

Let me now turn to the observations of the manuscript. While the authors use the corner charge description, demonstrating that would require breaking chiral symmetry which otherwise provides a much stronger protection.

In systems with chiral symmetry, a sufficiently strong disorder often results in a metallic phase, sometimes known as Gade-Wegner model.

The authors claim:
1. The appearance of a reentrant phase with corner states (the analog of corner charges with chiral symmetry).
2. The coexistence of this reentrant phase with gapless edge states.

These claims are both plausible and interesting. The first claim is plausible because periodic modulation of the hoppings results in Brillouin zone folding, and the appearance of minigaps. It is interesting because an alternative scenario would be that the metallic phase appears before gap reopening.
The second claim is plausible because:
- The edges have a different Hamiltonian than the bulk due to lattice termination.
- A disordered 1D system in this symmetry class may have a localized phase with or without a peak in the density of states or a critical point in conductance. A quasicrystalline phase is likely to inherit some of these properties.
- The closing of the edge gap does not change the topological properties because the bulk is an intrinsic higher order TI.

In summary, I believe the paper presents an interesting observation of a quasiperiodic topological phase, which may be used in follow-up research.

- Clarify that the result applies to the sublattice-symmetric system, a higher order TI. Alternatively demonstrates that the conclusions also hold with sublattice symmetry breaking, but this seems to be much harder.
- Fix the colorbar of the FIg. 1b to include the log in the label.
- Consider using a different colormap in the plots for easier readability (see e.g. https://bids.github.io/colormap/ for a further explanation)
- Consider including some of the context described in the report.

Ask for minor revision

1 - Find a new region in the phase diagram only existing at finite potential strengths
2 - Robust Methods to study the phenomenon

1 - Signals that determine the existence of the new region are very weak and subject to scrutiny

After the first round of corrections, the manuscript is significantly more presentable. It features a more complete exposition of concepts and tools and provides a more thorough overview of the literature.

However, I am still of the opinion that the shortcomings of the observations are still present, and I am not fully convinced that the results shown warrant the interpretation given. The conclusions would be much clearer if the authors would have considered a system that shows a much larger gap size, for instance.

Nonetheless, I think the paper should be published to ignite constructive scientific discussions.

None

Publish (meets expectations and criteria for this Journal)