Crystalline Weyl semimetal phase in Quantum Spin Hall systems under magnetic fields

1-The paper is well-structured and clearly written
2-The authors study two different models of a QSH insulator, thus making their results potentially applicable to a wider variety of systems
3-The authors study the effect of some PHS-breaking terms, which are expected to be present in real materials and break the protection of the topological phase

1-I believe that both the topological insulator and the topological semimetal phase are unstable against generic disorder. The latter should lead to a localization of edge and bulk modes, hindering the potential observation of this phase in a traditional, condensed-matter setup.

The authors study the effect of an in-plane Zeeman field on a QSH insulator, considering both the Kane-Mele and the Bernevig-Hughes-Zhang models. Although time-reversal symmetry is broken, the system remains topological due to the presence of particle-hole symmetry as well as mirror symmetry. Further upon increasing the magnetic field, the system evolves from a topological insulator to a trivial phase via an intermediate, topological semimetal phase, a scenario the authors refer to as an "unconventional topological phase transition."

The paper is well-structured and easy to follow. The authors characterize both the KM and BHZ models in a pedagogical, step-by-step manner, identifying the symmetries required for topological protection. Further, they also study the effect of PHS breaking terms, showing that for certain parameter regimes relevant to condensed-matter realizations of the QSHE, the topological insulator/semimetal phase may remain robust.

My main criticism of this work comes from the fact that I believe the topological insulator and semimetal phase to be unstable against disorder, hindering their potential observation in a conventional condensed-matter experiment. The models break TRS, so one would expect that generic disorder will lead to weak localization in two dimensions. I suspect that, even if the Rashba and NNN terms are not added, simply adding random disorder in the chemical potential and hopping terms will localize both the edge modes of the topological insulator and the bulk modes of the topological semimetal, rendering both phases trivial. This is in contrast to Ref. 37, where the topological semimetal is expected to remain conducting when weak disorder is present. This is due to the fact that TRS is intact in the model of Ref. 37, so weak anti-localization is possible in 2D.

I suggest that the authors add a section studying disorder in their system, showing, for instance, its effect on the conductance of edge and bulk modes. Further, the authors should comment more on potential realizations of their model in setups where the effects of disorder can be mitigated. These may include ultracold atoms or topoelectric circuits, which are mentioned only very briefly at the end of the paper.

1-Add a section studying the effect of disorder on the edge and bulk states of the topological insulator and the topological semimetal
2-Expand the discussion on meta-material realizations as potential ways of mitigating the effect of disorder
2-The authors introduce many abbreviations throughout the text, such as HSP, TRIM, RIM to refer to points in momentum space. The paper would be clearer if some of these are removed and the terms spelled out in full.
3-On page 7, I do not believe that the invariant characterizing the topological insulator should be called a mirror "Chern number." Chern numbers are integer and not Z2, and typically computed on even-dimensional (sub-)manifolds, whereas the authors consider 1D slices of the Brillouin zone. This terminology should be clarified in order to avoid confusion.

1-The authors give a concrete tight-binding model that exhibits a Weyl semimetal phase in two dimensions, protected by crystalline and particle-hole symmetries.
2-By contrast with previous work, this phase is stable against the breaking of time-reversal (T) symmetry, and is in fact induced by a T-breaking perturbation, an applied in-plane magnetic field.
3-The phase is moderately stable against some perturbations that break particle-hole symmetry.
4-The model considered can potentially describe real materials, such as HgTe/CdTe quantum wells and bismuthene on SiC.

1-The restriction to particle-hole symmetric Hamiltonians for a band insulator/semiconductor model (as opposed to a superconducting one) seems artificial.
2-Although small deviations from particle-hole symmetry are studied (Sec. 4 of the paper), it is not clear why the authors only consider the Rashba and next-nearest-neighbor hopping terms on the honeycomb lattice. Is there a specific reason why only these two terms are considered?
3-The magnetic fields required to observe the crystalline Weyl phase are such that the associated Zeeman energy would be on the order of the bandwidth, which would correspond to fields on the order of 10$^3$ - 10$^4$ T. Thus it appears very unlikely that this phase could be observed in practice.

The idea of applying an in-plane magnetic field to induce a 2D Weyl phase is interesting, and the authors do a good job of characterizing the Hamiltonian they are proposing, by studying the bulk/edge spectrum as well as computing topological invariants protected by various symmetries.

My main concern is how relevant this study is to real materials. Is there an advantage to realizing an unconventional TPT using an external magnetic field, as opposed to the inversion-breaking proposal of Ref. 37 where strain/electric fields are used? Could the authors elaborate on the idea in their last sentence, i.e., how the high magnetic field requirements could be circumvented by using topoelectrical circuits?

Also, I think the authors should better justify their choice of particle-hole symmetry breaking terms: are these terms energetically dominant for specific materials (or classes of materials)? For HgTe/CdTe, their Hamiltonian Eq. (10) does not contain bulk inversion asymmetry terms which are important for some of the physics of the quantum spin Hall state in this system. Do these affect their analysis from a symmetry point of view? Also, the simplest example of particle-hole symmetry breaking term, not discussed in the manuscript, is an on-site potential, as induced for example by random impurities. Can the authors comment on the effect of disorder on their conclusions regarding the stability of the crystalline 2D Weyl phase in their model?

A few minor issues:
- The authors use both "lattice constant $a$" and "step size $a$"; the latter is a bit confusing, and it would probably be best to use "lattice constant" throughout.
- Similarly, they sometimes use $i$ and sometimes $\iota$ for the imaginary unit.
- In the penultimate paragraph on p. 9, the first sentence contains a reference to Fig. 2(b), but I believe it should be Fig. 2(a) (?)
- p. 4, paragraph after Eq. (3): "magnetic field module"; the authors probably mean modulus (or magnitude, or amplitude)?
- Top of p. 12: "anysotropic" $\rightarrow$ "anisotropic"