A higher-order topological twist on cold-atom SO($5$) Dirac fields

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% Answer to the 2nd Referee
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We would like to thank the Referee for the assessment of our work, and for considering that it presents a compelling framework with broad multidisciplinary implications. Regarding the suggestions:


I) Comment: However, the manuscript falls short in suggesting methods for detecting key features of the model. Including such suggestions would significantly enhance the manuscript by bridging the theoretical framework with practical applications.

Answer: We thank the Referee for raising this point, which we have addressed in the amended version of the manuscript. We agree that a detailed discussion about the detection, including further numerical simulations for the specific model hereby studied, would be an additional value of the manuscript. However, we also stress that it would go beyond the original scope of this work, and make the manuscript even longer. Therefore, we have searched for a middle ground, addressing the definition of boundaries/corners and the detection and the in the new Sec. IV C. Here, we provide key references to quantum gas microscopes, and more recent results on the use of programmable optical potentials to create sharp boundaries in the optical lattices. We also comment on possible manifestations of the corner modes via adiabatic Thouless pumping, as well as the experimental inference of bulk entanglement by the reconstruction of the entanglement spectrum, as shown in recent works that are now cited in our amended text.

II) Comment: Moreover, the manuscript contains numerous spelling mistakes that warrant attention. I strongly encourage the authors to thoroughly review their work, paying particular attention to subtle errors that may be present in equations.

Answer: We agree with the Referee, as we have found additional spelling mistakes in the previous version of the manuscript. We have now checked this in detail, and are confident that the current version of the manuscript has now been improved.

We would like to thank the Referee for the assessment of our work.
Sincerely yours,
The authors

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We would like to thank the Referee for the very detailed report, and for making some specific recommendations that have helped us to clarify certain points. We were very pleased to read in the report that we managed to present our results with clarity and detail, and that these were indeed interesting an relevant not only for the study of HOTIs, but more generally for the research on quantum simulations of field theories. We now address in detail the points raised in the report:

I) Comment: The first concerns the "hidden SO(5) symmetry". Usually, speaking about an SO(5) symmetry, one refers to a model which is symmetric under the full group, as, for instance, the field theory in Eq. 4. At the beginning of section 5, instead, the authors write that: "We argued previously that the HOTI groundstate is protected by a hidden SO(5) symmetry and, thus, should be robust under symmetric perturbations". I think that the symmetries protecting the HOTI are discrete symmetries, and that the robustness of the HOTI phase is much stronger. In particular, I tend to think that perturbations which are SO(5) symmetric are very artificial and not realistic in the ultracold atom system on the lattice. Indeed the HOTI phase of the non-interacting Hamiltonian 29 does not require a full SO(5) symmetry. Therefore I find the previous statement misleading. 
In general, I do not think the SO(5) symmetry plays an important role in defining the HOTI phase, as also showed in Fig.4, but only a small subset of discrete symmetries. A similar statement appears in the abstract, and, again, in my opinion it is confusing. And I find it confusing to refer to "a hidden SO(5) symmetry" (as in the abstract) because, if I understand correctly, this symmetry is just a discrete Pi/2 rotation in real space complemented with the specific transformation in Eq. 35, without any of the complexity of the continuous group. Therefore I would refrain from speaking about a "hidden SO(5) symmetry" to refer just to a specific rotation.

Answer: We thank the Referee for this detailed comment, and for explaining the reasons that can potentially raise a misunderstanding when using the words ‘hidden SO(5) symmetry’ in the text. We fully agree that the relevant symmetry for the HOTi and all of the underlying physics discussed in this work is a discrete SO(5) rotation, not the full continuous symmetry group. The reason why we used ‘hidden SO(5) symmetry’ comes from the fact that, in the cold-atom community, there are previous works that focus on the appearance of this symmetry in spin-3/2 ultra cold atomic gases, as we have discussed and referenced in the text. However, it is true that, when including the additional terms that are responsible for the HOTI physics, the continuous symmetry is explicitly broken, and it is only a discrete SO(5) rotation that survives and is responsible for the symmetry protection of the HOTi phase.

In order to avoid any possible misunderstandings, we have restrained from using ‘hidden SO(5) symmetry’ in the amended version of the manuscript. We have substituted ‘hidden SO(5) symmetry’ by ‘discrete SO(5) rotation’ everywhere in the hope that it will not throw the potential readers into confusion. We thank the Referee for raising this important point.

II) Comment: A second important point, in my opinion, is related to the experimental proposal. The paper lacks any concrete proposal about how to detect the HOTI phase in experiments or how to observe the main features of the model. I think this is a fundamental point to address in detail. It is crucial, for an experimental proposal as the one presented, to discuss how to validate its results. Additionally, I also think that another important and related element that has not been discussed is how to obtain well-defined surfaces and corners in the optical lattice, and to discuss, for instance, what can happen to the corner topological modes in cases in which sharp surfaces are replaced, instead, by position-dependent confining potentials.

Answer: We thank the Referee for raising this point, which we have addressed in the amended version of the manuscript. We agree that a detailed discussion about the detection, including further numerical simulations for the specific model hereby studied, would be an additional value of the manuscript. However, we also stress that it would go beyond the original scope of this work, and make the manuscript even longer. Therefore, we have searched for a middle ground, addressing the definition of boundaries/corners and the detection and the in the new Sec. IV C. Here, we provide key references to quantum gas microscopes, and more recent results on the use of programmable optical potentials to create sharp boundaries in the optical lattices. We also comment on possible manifestations of the corner modes via adiabatic Thouless pumping, as well as the experimental inference of bulk entanglement by the reconstruction of the entanglement spectrum, as shown in recent works that are now cited in our amended text.

We now address the additional minor points also raised by the Referee: 1) If the authors wish, they could mention that also Weyl semimetals offer a potential example of 3D systems whose low-energy physics is well captured by relativistic (but not interacting) Dirac fields. Answer: We have introduced a sentence in the introduction, together with a couple of references. 2) t is used both for time, and the tunneling amplitudes in Eqs. 12 and 13. I suggest changing the font or the notation. is used both for time, and the tunneling amplitudes in Eqs. 12 and 13. I suggest changing the font or the notation. Answer: We have introduced.a sentence below Eq. (13) to avoid this possible confusion, emphasizing that only Euclidean time \tau will appear later on. 
3) In the definition of U0 and UFt, what is k ? I am a bit puzzled because the densities are taken in real space on the lattice, and I find it misleading having an onsite density-density interaction that depends on momentum. Answer: The $k$ that appears in both expressions is the laser wave vector. We have added a subscript $k_L$ to avoid any possible confusion. 
4) The authors write: "At the level of the twisted-mass free Hamiltonian (11), one sees that βH0(k)β=−H0(k). This corresponds to the AIII class in the classification of topological insulators". I tend to disagree with this statement. When we consider the Hamiltonian 29, we see that, besides this sublattice chiral symmetry, the model displays also a particle-hole symmetry given by σz⊗σx and a time-reversal symmetry given by σx⊗σx (following the standard classification of topological insulators ans superconductors). Therefore I would say that the topological class is BDI instead of AIII. Furthermore, for the sake of clarity, I would consider these as non-spacial symmetries rather than global symmetries, to avoid confusion with the unitary global symmetries discussed in the work. Answer: We thank the Referee for bringing this comment. We have changed our notation, and now refer to ’spatial and non-spatial symmetries’, as suggested. Regrading the first point, we thank the referee for noticing this mistake. We had been previously working with a different discretization in which $sin(k_ia) \leftrightarrow cos(k_ia)$, such that the explicit particle-hole and time-reversal symmetry where broken unless the bare masses $\tilde{m}_1=\tilde{m}_2=0$. However, for the current choice of the manuscript, the Referee is correct that this should correspond to a BDI class with C=σz⊗σxComplex_conjugation and T=σx⊗σxComplex_conjugation. We have corrected this on the amended manuscript. The referee has also spotted the following typos page 5: "lea to" -> "lead to"
page 5: tau -> \tau
caption of Fig. 1: "decouple"->"decoupled"
page 9: "micorsocopic"->"microscopic"
page 11: "of second order"->"of second order"
page 14: "pseudo-sclarar"->"pseudo-scalar"
page 17: "the higher-order topological invariant an the corner modes cannot coexist"
page 18: "spin-3/3 Fermi gases"
page 18: "it woudl" Which we have corrected. We would like to thank the Referee for all the helpful suggestions, which have helped us to improve the presentation of our results, and correct possible sources of confusion. Sincerely yours, The authors