THz Higher-Order Topological Photonics in Ge-on-Si Heterostructures

This paper studies the interesting topic of topological phases in 2D Photonic Crystals. Both the architecture of the crystals and the methodology have been previously introduced in the literature (Ref 37). The paper rather focuses on the system's behavior regarding a change of some geometrical aspects. Specifically, it measures , through numerical means, the band gap variation as a function of the ratio of the size of the Ge stand to the lattice cell, as demonstrated in Figure 1. One of the main results of the papers is the claim of the presence of a topological phase transition.

Despite the paper's elegant presentation, I have several concerns:

1. The authors state that compressed and expanded PC belong to distinct topological phases, with the bands' parity serving as the topological invariant. However, the arguments presented are not convincing. The two configurations are related by a translation (a/2,a/2) and one can notice from figure 2 that wavefunctions of the electromagnetic field could simply be related by the same translation. both for the lower and upper bands. So, the change of parity seems trivially a result of the change of reference. It would be helpful to clarify this.

2. The paper states that the clearest indicator of a topological transition is the appearance of spatially confined guided modes at the boundary between domains of differing band topology, citing references [5, 7, 56–58]. However, upon reviewing reference 5, I found no discussion of this nature. It would be constructive for the reader to provide a precise reference that for such discussion.

3. Unlike reference 37, this study lacks experimental data comparison, which necessitates at least an examination of the numerical methods' fidelity. Details such as error estimation, mesh sensitivity, and convergence rate, along with parameters selected for the FEM simulations, would aid in reproducibility and validation of the numerical findings.

4. Given the scalability of Photonic Crystals, any frequency range could be achieved with a change of scale. It is unclear why the THz range is specifically singled out. Including it in the title seems somewhat contrived to me.

5. References to "supplementary materials" on page 3 were noted, yet I couldn't find any of these supplementary materials in the submission.

6. It would be interesting to mention the role of the edge (111) of the Germanium stand and what would be the behavior of the crystal in its absence.

7. Minor typographical errors identified include "perfomed" which should be corrected to "performed," and "simmetry" to "symmetry."

Address remarks in the report