Engineering a pure Dirac regime in ZrTe$_5$

The authors propose a method (strain) to obtain a pure Dirac phase.
The measurements and calculations are sound, the material studied is interesting, and the search for pure Dirac phases is indeed timely. The combination of synchrotron diffraction experiments and ab-initio calculation is a strong point of the work, although the ab-initio calculations are only performed at the DFT level.

My main criticism is about the robustness of the pure Dirac phase found with this method (see report).

If the goal is to find a Dirac semimetal (i.e. Delta_Gamma = 0) without unwanted additional pockets, I think they should be looking for something more robust than just the Dirac-like gap closing of the phase transition. Otherwise, finding a sample that actually presents such a Dirac point would be just a matter of luck of having the sample at exactly the right lattice parameters. A tiny difference in the parameters would break the Dirac crossing, getting the sample out of the Dirac phase.

In other words, where is Delta_Gamma = 0 in the phase space of Fig. 5? This must be only a point (or a discrete line? surprisingly, they don't show Delta_Gamma in Fig. 5) making it almost impossible to tune the sample to land exactly on that particular interesting point. And if this is achieved, any tiny distortion will move the sample outside that spot. This is not a problem of the material, but rather of the method proposed in the paper. Any search for Dirac physics based on the Dirac gap closing in a topological phase transition is not practical for real applications.

What would be really interesting is to find a way to remove the trivial pockets without opening a gap at the Dirac point. Unfortunately, the method proposed here does both. Instead one needs to look for a Dirac crossing that is robust and protected by some symmetry. In other words, there should be a finite region (not discrete) in the phase space of Fig. 5 where Delta_Gamma is zero.

I am sure the paper deserves publication in some form, but I am unsure about the impact of the work for publication in SciPost Physics. Since this robustness is important for applications and this is key for the impact of the work, I think the authors should comment on the robustness of this phase and why it is interesting nevertheless, before it can be published.

I have a couple of other comments:

- The authors do comment on the limitations of GGA for the prediction of the band structure. It would be good to do something beyond to see the extend of the many-body effects in this compound. It should not be a huge effort to calculate the band structure (of the unstrained material) with a method beyond GGA (a hybrid functional, for example).

- Other points that lower the impact of the work are that 1) the topological transition upon strain is already known (as they correctly cite) and the only thing added here is the effect on the trivial pockets. And 2) the fact that strain can move pockets up and down in energy is in itself not very novel.

- Figure 3: In the orange curve there seems to be a hole pocket but no electron pocket that compensates for it. In the blue curve, the whole pocket seems much larger than the electron pocket. Is the electron pocket that compensates the hole pocket somewhere else (not shown) within the Brillouin zone or is this a problem of convergence of the Fermi energy?

- About this sentence:
"While a Dirac-like gap closing is guaranteed at the topological phase transition, we here aim at identifying an extended parameter regime without unwanted additional pockets close-by in energy, and in which there is a direct gap close to the Gamma-point in reciprocal space."
Why do they want to have a "direct gap close to the Gamma-point" and what do they mean with that?

- Other computational parameters of the DFT calculations should be provided (not just the k meshes).

- It seems like they use the term "pockets" for all the minima of the conduction band, even when they are relatively far from the Fermi energy. I wonder if this notation is appropriate. This minima would not appear as pockets in a plot of the Fermi surface, so I am not sure if one should call them "pockets". For example, Figure 3(a) is a semiconductor, not a semimetal, so speaking about "pockets" seems odd.

- The paper is well structured, easy to follow the results and their discussion.
- Timely subject matter, discussing the tuning of topological properties of ZrTe5 by externally applied mechanical deformation.

- The DFT code the authors use is not state of the art, as the authors themselves point out in the conclusions

ZrTe5 is well known to be at the boundary of a topological phase transition, where weak and strong topological insulator and Dirac semimetal phases are relatively easily achieved by tuning the lattice parameters. The paper proposes a method to strain engineer the single particle band structure of ZrTe5, in order to realize a 3D Dirac semimetal phase. This sensitivity to relatively small changes in lattice parameters makes this material interesting, since it promises the realization of topological phase transitions under strains available in laboratory conditions. This is a very timely topic, as such it qualifies for inclusion in SciPost Physics.

- It would be worthwhile to discuss the results in light of other 3D Dirac semimetals, such as Na3Bi and Cd3As2, see for example:
Collins, J. L. et al. Electric-field-tuned topological phase transition in ultrathin Na3Bi. Nature 564, 390–394 (2018)
Jeon, S. et al. Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2. Nat. Mater. 13, 851–856 (2014)
Is the Dirac band also masked by other trivial bands in these cases? It seems to me that they are not. It would be to the benefit of the paper, if the authors expanded with some concrete examples on their statement in the introduction: “Most of the time, the Dirac points of 3D crystals occur at energies away from the Fermi level, or they are surrounded by additional trivial electronic pockets which obscure the unique features of linearly dispersing electrons”. Furthermore, in the case of thin Na3Bi, tuning of the topological phase by outside parameters (electric field in the case of the Jeon etal. results) is also possible, making it relevant to the current work. It might be fruitful to just mention other methods of tuning topological phases, outside of mechanical strain.
- Why is there a difference in the van der Waals gap in the c-direction in the two samples? What could be the reason for this? Maybe disorder, changes in stoichiometry?
- The assumption that the Poisson ratio in the ab and ac directions is the same seems dubious to me, since the crystal is clearly anisotropic in these directions. Why is this a valid assumption? How robust are the conclusions of the paper based on this assumption? For example, how much would a 1% and a 10% difference in Poisson ratio influence the topological phase transition?

In the manuscript entitled « Engineering a pure Dirac regime in ZrTe5 » J. I. Facio and coworkers report an ab initio investigation of the band structure of ZrTe5. First, the authors have carefully determined the crystal structure and atomic positions by means of XRD. Second, the authors computed the band structure corresponding to different atomic positions and under strain. The core of the work is to isolate the parameters that allow not only the Dirac phase (closure of the direct gap at Γ) but also the absence of additional, and trivial, pocket at the Fermi level. Hence, this work deals with an important topic, the realization of a real Dirac semimetal phase where new exotic transport properties and quantum effects might be under the reach of experiments. I have found the manuscript well organized and written, and the conclusions sound solid.
I have nonetheless few questions that I kindly ask the authors to consider.

1) The authors propose the van der Waals gap rc to be “the most” important factor (maybe one important factor would be proper) determining the Dirac nature. In order to modify it, the authors play with the hopping integral between the Te atoms across the van der Waals gap. However, this parameter seems difficult to be controlled from an experimental point of view. Intercalation of alkali metal is, in my opinion a viable route, compatible at least with surface sensitive technique such as STS and ARPES. Can the authors propose alternative methods to experimentally confirm their prediction?
2) As the authors point out, ZrTe5 crystals of structure A and B display different temperature of the resistivity anomaly. Besides the structural differences, those crystals are synthetized via different methods, (Te flux, CVT) and they exhibit different defects. The importance of defects has been experimentally explored by B. Salzmann et al in Phys. Rev. Mater. 4 114201 (2020). Have the authors considered the role of defects, directly in the calculations and their influence in van der Waals gap rc ?
3) In the conclusions the authors leave the problem of electronic correlations, and the influence on the band gap, unsolved. Can the authors indicate the best strategies they would follow to tackle this problem?
4) In my opinion a few important additional references are missing. The evolution of the electronic properties of ZrTe5 with uniaxial strain has been experimentally investigated by P. Zhang et al., Nat. Commun. 12 406 (2021). The temperature evolution of the band structure and of the chemical potential have been first reported by G. Manzoni et al, Phys. Rev. Lett. 115 207402 (2015)

A) Minor: in the introduction the authors say “…powerful tool for in-situ tuning of the band structure such that all nontrivial pockets are pushed away from the Fermi energy…”. I guess the aim is to push the trivial pocket far from the Fermi energy, thus leaving only the topological one to be responsible for the conduction.