Topological hydrodynamic modes and holography

Submission summary

 As Contributors: Ya Wen Sun Preprint link: scipost_202011_00004v2 Date submitted: 2021-02-17 14:18 Submitted by: Sun, Ya Wen Submitted to: SciPost Physics Academic field: Physics Specialties: Gravitation, Cosmology and Astroparticle Physics Approach: Theoretical

Abstract

We study topological gapless modes in relativistic hydrodynamics by weakly breaking the conservation of energy momentum tensor. Several systems have been found to have topologically nontrivial crossing nodes in the spectrum of hydrodynamic modes and some of them are only topologically nontrivial with the protection of certain spacetime symmetries. The nontrivial topology for all these systems is further confirmed from the existence of undetermined Berry phases. Associated transport properties and second order effects have also been studied for these systems. The non-conservation terms of the energy momentum tensor could come from an external effective symmetric tensor matter field or a gravitational field which could be generated by a specific coordinate transformation from the flat spacetime. Finally we introduce a possible holographic realization of one of these systems. We propose a new method to calculate the holographic Ward identities for the energy momentum tensor without calculating out all components of the Green functions and match the Ward identities of both sides.

Current status:
Editor-in-charge assigned

Author comments upon resubmission

Dear Editor,

We would like to thank both referees for useful suggestions which help us improve our manuscript! We could see that all the questions from both referees are due to the unfamiliar terminology in topological states of matter so we expanded our manuscript and added a lot of more detailed explanations to the text. In the following we will first reply to Referee A’s report and list our modifications accordingly under the tab of “list of changes”. Then we reply to Referee B’s report and list our modifications accordingly also under the tab of “list of changes”.

Reply to Referee A and modifications

We would like to thank Referee A for very helpful questions and suggestions to help us improve our manuscript. In the following we will first answer Referee A’s questions and point out the modifications that we made in our manuscript accordingly. To be more clear, we have denoted all the modifications to Referee A’s comments in the color blue in the manuscript.

Question 1. It is not quite clear to me why we should identify a band crossing as a gapless mode (as after all these do not have zero energy in any sense) and Requested change 3. Further discussion on why the existence of band-crossing points should be viewed as analogous to \omega = 0 in more traditional examples of physics.

Similar to what happens in condensed matter physics, whether the bandcrossing is at w=0 is not important and the important thing is that it is a real bandcrossing instead of an accidental touching. As long as two bands cross instead of not touching at all, there will be no gaps in the spectrum so we could all it gapless. We are interested in whether the bandcrossing nodes will become gapped while it does not matter whether it is at w=0. Also we usually study effective excitations near each of the band crossing points so the energy at the bandcrossings could each be set to zero effectively. This is what people usually do in condensed matter physics, i.e. they usually study the effective Hamiltonian near the crossing nodes so the energy at the crossing point is set to 0. More detailed explanation could be found in the blue paragraph at the end of page 10 that we added in the new version.

Question 2.1 It remains a bit unclear what exactly is topologically protected.

We have expanded the definition of a topologically protected gapless state on the upper half of page 11 in blue. We also copy the paragraph here in the following. We also need to emphasize here that the definition of a topologically nontrivial gapless state is a state which does not have any gap at any ω and under small perturbations would not develop a gap. This is because as long as a gapless state could develop a gap and becomes a trivial gapped state under small perturbations, this would mean that the gapless state is topologically equivalent to the trivial vacuum. Thus for a topologically nontrivial gapless state, by being topologically protected, we refer to the fact that the state remains gapless under small perturbations of the system which usually could gap a system. This indicates that the bandcrossing points should be singular points in the momentum space. Being topologically inequivalent to the trivial vacuum, this topologically nontrivial gapless state should possess a nontrivial topological invariant which takes a different value compared to the trivial vacuum, e.g. the two nodes of a Weyl semimetal possess nontrivial chiral charges which are different from the trivial value of zero thus the two nodes cannot be gapped under small perturbations of the system. The topology of states depends on the Hamiltonian or equivalently the wave functions and is an intrinsic property of the Hamiltonian. Regarding this point, Figure 1 from arXiv. 1609.05414 “Topological nodal line semimetals” by Chen Fang, et.al. shows very clearly what accidental and topologically protected bandcrossings are.

Only (a) and (c) in Figure 1 of arXiv. 1609.05414 are needed here. (a) is an accidental band crossing and (c) is a topologically protected band crossing where different colors could be viewed as different bands. In (a) a small perturbation could gap the nodes and in (c) a small perturbation only changes the distance between different nodes.

Question 2.2. The second point is addressed in section 5, where a potential topological invariant is discussed. Perhaps I am confused by the low-dimensionality, but I do not fully understand why the fact that the two degenerate states are orthogonal means that there is a topological protection. Indeed it is shown explicitly in Figure 3 that turning on a small ky is sufficient to connect the states. and Requested change 1. Further explanation of the topological invariant.

In the new version we have explained in more detail about how the topological invariant works in our simple case, i.e. why the two adjacent states being orthogonal means that the system is topologically nontrivial. The explicit places include the middle of page 11 “Being topologically inequivalent to the trivial vacuum, this topologically nontrivial gapless state should possess a nontrivial topological invariant which takes a different value compared to the trivial vacuum”, and the blue words in the first two pages of section 5 on page 25.

Regarding “turning on a small ky is sufficient to connect the states”, we added a very detailed explanation in the last blue paragraph of page 11 that a seeming gap by turning on a ky is not a gap and this only states that the crossing nodes are points in the momentum space while not lines or surfaces. More details could be found in that paragraph.

Question 3. As far as I understand the usual state of affairs in Weyl semimetals is that there are two gapless modes which are separated in momentum space, and can only gap out when they join and annihilate. Here the two modes always touch in momentum space, and it seems they can gap out at any time. The authors state that the perturbation which can cause the gapping out is protected by a symmetry, but the symmetry in question is somewhat confusing to me, as it is not a restriction on the theory but rather on the space of solutions to the theory, i.e. setting kx = 0). and Requested change 2. Further explanation of the nature of the symmetry that protects the dispersion relation.

In our case the two modes could not be gapped out by the mass term in the x direction but they could be gapped out by the mass term in the y and z directions. This means that they are topologically protected only when the mass terms in the y and z directions are forbidden. In this case it is called a symmetry protected topological state for which we require a symmetry that forbids the y and z direction mass terms, i.e. the y and z direction mass terms do not obey this symmetry. It is a restriction on the theory (on the possible deformations of the effective Hamiltonian) but not on the solutions: as we explained in point 2.2, this is not related to setting ky and kz to be zero. However, setting ky and kz to be zero is a way to calculate the topological invariant in this case as it is the high symmetric point which is used to calculate the topological invariant of a symmetry protected topological state.

We have added these explanations in the text, including the last two paragraphs on page 11 and the beginning of page 27.

Regarding questions 2.2 and 3, Figure 2 in arXiv. 1609.05414 “Topological nodal line semimetals” by Chen Fang, et.al. is extremely helpful in understanding.

In all three cases in Figure 2 of arXiv. 1609.05414, the nodal points form a circle denoted in black in the figure. This figure shows the calculation of the topological invariant for the mirror symmetry protected case (a), the normal case without a protection of a symmetry (b), and for (c) they want to know whether the whole circle is topologically nontrivial if the whole circle shrinks to a point, i.e. if the point is topologically protected. The topological invariant for (b) is calculated along a circle (the circle lives in an effectively two dimensional space, but the circle is one dimensional). The topological invariant for (c) is calculated on the surface of a sphere (the sphere lives in an effectively three dimensional space, but the surface is two dimensional). This case is very similar to the calculation of the topological invariant for a Weyl semimetal, just to replace the inner circle with a Weyl point. For (a), the nodal line is symmetry protected by the mirror symmetry as stated in the caption of the figure. Thus the topological invariant has to be calculated on the high symmetric plane, i.e. the mirror plane: the blue plane. To calculate the topological invariant in (a), we only need to compare the two red points on the blue plane and the two points live in effective one dimensional space (but the points are zero dimensional). This is exactly what happens in our case.

Question 4. Furthermore, I confess that the discussion in Section 4.2.2 gives me further cause for doubt; after all, that is simply conventional (topologically trivial) hydrodynamics in a different coordinate system, and I feel that a fundamental property regarding whether or not a mode is topologically protected should not depend on the coordinates used to describe it. Our answer: The referee is correct that our system is not in a different coordinate system but in a different reference frame. We have not explicitly distinguished between the “coordinate system” and the “reference frame” as usually people could tell from the context. However, here apparently it causes confusion. We would like to thank the referee for pointing this out and we have changed all the words “coordinate transformation” that in fact mean reference frame transformation in our manuscript to the words “reference frame transformation”.

Reply to Referee B

Our reply to Referee B was already posted to the manuscript page one day after the report arrived. We also append the reply here in the following.

We would like to thank the referee's comments! The referee's report showed that we need to add more explanations in our manuscript on those points which are basic knowledge in condensed matter physics but are not familiar to the high energy community. We will answer the referee's comments point by point as follows and we will improve our manuscript in the next revised version.

In the following, we first list the referee's question or comment and then answer after each point.

1) However, in the absence of any microscopic discussions and the reasons for this Ward identity, how do we know that \Gamma isn’t imaginary? In that case, they would find a real contribution to the spectrum. How would this be related to any kind of "topology”?

Our answer: The referee asked a question about appendix A. The question is if \Gamma in appendix A could also be imaginary so that the spectrum could also be real. Then the referee asked if their spectrum is real, how would this related to any kind of topology. We need to emphasize that Appendix A has nothing to do with our system in this manuscript. We added appendix A only for readers not to mix our system with the momentum dissipation systems which have been studied a lot in holography. No matter whether their spectrum could be real, that has no relation to topology, at least not any that we could directly see. Our system has a nontrivial topological structure not because their spectrum is real, i.e. we are not trying to find a real spectrum in hydrodynamics and claim that this is topologically nontrivial because it has real spectrum. Real spectrum is a very basic and necessary requirement for us to analyze the topological structure of the system but it does not mean that as long as we have real spectrum, we would have a nontrivial topological structure. Whether the system has a nontrivial topological structure depends on the structure of the effective Hamiltonian or equivalently the structure of the spectrum. In the system of Appendix A, momentum dissipation could be caused by external fields as well as broken diffeomorphism invariance and in all cases that have been studied \Gamma is real while not imaginary. In summary, appendix A is not related to our system. Their \Gamma is real and even if it could be imaginary, it does not guarantee a nontrivial topological structure.

2) This latter point is the main source of my discomfort with this paper. The modifications of the Ward identity seem quite arbitrary so the results can be anything. And yes, they are tuned to give a real gap but I don’t see anything fundamentally revealing about that.

Our answer: By modifications of the Ward identity, the referee refers to the modification to the energy momentum conservation equation. Sure modifications could be arbitrary but only a specific modification could lead to a nontrivial topological structure: the one in our manuscript. More importantly, we need to emphasize that our system is a topologically nontrivial gapless state, not a gapped one. Our modification does not give a real gap but gives nontrivial bandcrossings as shown in figure 3 in the manuscript. We would explain why this is topologically nontrivial below. Before that we emphasize the revealing point of our work: we found that a specific modification to the energy-momentum conservation equation leads to non-trivial topological gapless states. This specific modification could come from a change of reference frame from an inertial one to a non-inertial one, which indicates that an accelerating observer could observe topologically nontrivial modes.

3) Why does this have to imply (or rather, be a consequence of) non-trivial topology? I do not think that this is explained in this work.

Our answer: This is not a consequence of non-trivial topology, so the first part is the correct question, i.e. this implies non-trivial topology. Let us answer why this implies non-trivial topology. This modification to the energy momentum conservation leads to a special structure of spectrum. From the point of the construction of an effective Hamiltonian with the modification, this already reduces to a condensed matter problem of topological states of matter. Our system is a very typical topological gapless state in condensed matter physics. Just according to the topological band theory, gapless topological states of matter are those that have band crossings in their spectrum which cannot be gapped by small perturbations, which means that the band crossings are not accidental crossings. A review I often refer to is arxiv. 1609.05414. In their figure 1, you would find accidental (topologically trivial) gapless state and topologically non-trivial gapless state. In our system, we can show that the band crossings will not be gapped by small perturbations that obey a certain symmetry. Here there is also some physics about symmetry protected topological states and if the referee needs this information, we could explain more in a next reply. We also calculated the topological invariant in this case. The so called undermined Berry phase is in fact the one spatial dimensional analog of the three spatial dimensional Berry curvature and two spatial dimensional Berry phase, i.e. the one spatial dimensional topological invariant. In one spatial dimension, it is super simple as there are only two points that we need to check, see case a) in figure 2 of arxiv. 1609.05414. Undermined Berry phase means that the two states are orthogonal to each other so they cannot be deformed to each other by small perturbations. When this topological invariant is different from the value for the vacuum (which is a topologically trivial state), the system would be in a topologically nontrivial state. Regarding the second sentence in this question, we have explained this in our work, e.g. in the introduction we mentioned “Besides the fact that the crossing nodes in these systems will not become gapped under small perturbations, we will also show evidence of the nontrivial topology from the existence of undetermined Berry phases. “, and also in section 5. Indeed when we check where we mentioned these, we found that because these are basic knowledge in condensed matter physics, we did not explain them in detail and only mentioned the conclusion in one or two places, which is surely a barrier for high energy theory people, so we will improve this in our next version. Thanks to the referee for pointing this out.

4) I also do not find the discussion of the "undetermined Berry phases’’ convincing enough to be sure that this is related to the existence of a real gap and related to their hydrodynamic considerations.

Our answer: As we emphasized above, our system is a gapless topological state, where the gap in our paper in section 3.2 refers to the gap term that can gap the standard hydrodynamic system but cannot gap our system. We need this gap term to show that our system is still gapless with this term present. As we explained above, our system is a typical gapless topological state in the topological band theory of condensed matter physics. This undetermined Berry phase is a topological invariant that characterizes if our state is of the same topology as the vacuum. In this part, as long as we have the same effective Hamiltonian in eqn. (3.7) (3.8), the topological structure would be the same, no matter whether it comes from a hydrodynamic system or an electronic system or even an optical system. Of course here it comes from our relativistic hydrodynamic system. Thus it reveals that there is a topological structure in the spectrum of our hydrodynamic system.

5) Can these claims be made precise by significantly expanding what is at present written e.g. in Section 5? Can one really derive’’ their Ward identities directly by some topological considerations and show that these Ward identities cannot be "derived/constructed’’ in another way (beyond noticing analogies)?

Our answer: To explain the second equation, the ''modifications to the Ward identities” lead to a specific spectrum that has a nontrivial topological structure. This has the same logic as the fact that an addition of a time reversal symmetry breaking term in the Dirac equation would lead to a Weyl semimetal spectrum with a nontrivial topological structure as could be found in e.g our paper Phys.Rev.Lett. 116 (2016) 8, 081602, arxiv.1511.05505 . The question “ Can one really derive their Ward identities directly by some topological considerations and show that these Ward identities cannot be "derived/constructed’’ in another way (beyond noticing analogies)?” is the same as asking "can one derive the Weyl Hamiltonian from their topological structure?”. The answer is that the topological structure is a consequence of the structure of the Hamiltonian and we could only check if each Hamiltonian would lead to a nontrivial topological structure, while not the inverse way. Thus we cannot derive the Hamiltonian directly from the topological structure and there is no need to do so either. We will explain more in section 5 adding the explanations above and thanks to the referee for pointing this out. Finally, we emphasize that our work has no relation to appendix A which is why we added Appendix A in order to avoid any confusion to mix the two different systems. We hope this clarifies the confusion from the referee and we would be happy to discuss and communicate more whenever the referee has any new confusions or questions.

We hope this expanded version of the manuscript is more clear and could be published in SciPost in its current form.

Yours sincerely, Yan Liu and Ya-Wen Sun

List of changes

List of changes:

All the blue text in the revised manuscript are modifications following Referee A's suggestions and all the red text are modifications following Referee B's suggestions.

For modifications regarding referee A's suggestions, we have mentioned each of them in the corresponding questions in the reply to referee A above.
For modifications regarding referee B's suggestions, we list them in the following separately.

We made modifications according to Referee B’s questions in the text using the red color. As we already stated in the reply to Referee B’s questions, all the questions from Referee B are due to misunderstanding caused by unfamiliar condensed matter terminology on topological states of matter. Thus we expanded the explanations in our text to make the manuscript more friendly to high-energy readers who are not familiar with topological states of matter. We would like to thank Referee B for helpful questions that help us improve our manuscript.

We denote the points of the modifications also using the numbering of the questions in our reply to Referee B. All the questions from Referee B are answered in detail in the reply above, and we made modifications in the text accordingly in necessary places.

1 Regarding point 1 from Referee B on appendix A, we added a short paragraph at the end of page 12 and the beginning of page 13 in the manuscript explaining while the momentum dissipation system in Appendix A has nothing to do with our work or with topological states of matter.

2 Regarding point 2 from Referee B on the modifications to the energy-momentum conservation equation, we have explained clearly in the reply that the modifications are not arbitrary but very carefully chosen. We then added a short emphasis on this point at the beginning of section 3.3 on the middle page. 8.

3 Regarding point 3 from Referee B on why we have nontrivial topology in the system. Besides the detailed answer in the reply, we have expanded the explanation in the manuscript. Now the blue and black sentences on page 11 give a more detailed explanation on this point.

4 Regarding point 4 from Referee B on the topological invariant, besides the explanations in the reply, we have expanded in our manuscript and the first two pages in section 5 give a more detailed description of the topological invariant in one effective spatial dimension.

As the referee mentioned “gap” in points 2 and 4 when our system is not gapped but gapless, at the end of page 7 we added a short explanation on what role the “gap” plays in our gapless topological states.

5 Regarding point 5 from Referee B, besides the answer in the reply, we also emphasized on page 26 in red on the relationship between nontrivial topological and the modification to the conservation equation.

Submission & Refereeing History

Resubmission scipost_202011_00004v2 on 17 February 2021
Submission scipost_202011_00004v1 on 5 November 2020

Reports on this Submission

Report

I have focused my attention on the topological analysis of the paper entitled "Topological hydrodynamic modes and holography", and I must say that I am quite sceptical. I am confused by their argument that mix both Chern numbers (which are indeed a good topological number to characterize Weyl nodes) and Berry phase. In graphene, the quantized Berry (= \pm pi) and is said to be topological for that reason. So the statement of the authors saying that a " node would possess a nontrivial topological invariant when an undetermined Berry phase exists" is doubtful.

I am also skeptical about the following procedure "Here as we are in one spatial dimension effectively and the manifold enclosing the node is in zero residual spatial dimension, the calculation of the topological invariant is different from the Berry phase or Berry curvature for nodal line or Weyl semimetals". I don't understand what topological number the authors get out from this procedure.
I fear that by reducing their 4D problem to a 1D one, they miss sound topological interpretations.

Usually, in band insulators, topological indices can be computed locally from high symmetry points of the Brillouin zone, where the sign of some relevant quantity is evaluated (see e.g. Z2 topological index in 3D time-reversal invariant topological insulators with inversion symmetry).
Another example, maybe closer to what is discussed in that manuscript, would consist in looking at the Chern numbers associated to each band crossing in 3D parameter space instead of 1D parameter space, i.e. by fixing one of their 4 dimensions.
Another procedure to characterize band crossing points is by the homotopy properties of the map k -> h(k), which in Weyl nodes H=h(k).sigma , is encoded into the homotopy group of the sphere \pi_2(S^2).

I can understand that one of those procedures would yield a well-defined (even though maybe not sufficient) topological characterization of their band crossing points. But in the present form, I must say that I am not convinced by the topological characterization of those nodes. They probably are, but, in my opinion, this remains to be shown.

• validity: low
• significance: -
• originality: -
• clarity: low
• formatting: -
• grammar: -

Author Ya Wen Sun on 2021-05-05
(in reply to Report 3 on 2021-05-05)

We would like to thank the referee for useful comments and suggestions. We will reply to the referee's report sentence by sentence in the following. As we have shown that our system is indeed topologically nontrivial both from the definition of a nontrivial topological invariant and from the robustness of the system from small perturbations of the system, we believe our work is complete and solid. However, we would nevertheless be happy to discuss more with the referee in order to seek other possible definitions of topological invariants in our case, e.g. generalizing other possible zero dimensional topological invariants to our case with or without protecting symmetries.

1. The referee said "I am confused by their argument that mix both Chern numbers (which are indeed a good topological number to characterize Weyl nodes) and Berry phase. In graphene, the quantized Berry (= \pm pi) and is said to be topological for that reason. So the statement of the authors saying that a " node would possess a nontrivial topological invariant when an undetermined Berry phase exists" is doubtful."

We did not mix the Chern numbers and the Berry phase. As we explained in the manuscript and in the reply to the other two referees, Chern numbers (calculated from the integration of the Berry curvature) are for the Weyl nodes, which are calculated on a sphere, i.e. an effectively two dimensional surface enclosing the node. Berry phase is used when the surface" enclosing the nodes is effectively one dimensional, e.g. for normal nodal line semimetals in three spatial dimensions or for two dimensional materials as the referee stated. They differ because of different spatial dimensions. We would like to refer the referee to Fig. 2 of arXiv. 1609.05414. (Sorry that we cannot append a picture in this reply form.) In Fig.2, the Weyl semimetal case could be the circle shrinking to point limit of c) in that figure. The nodal line is the second b) case. Our case as emphasized corresponds to the effectively zero dimensional case, i.e. case a) in Fig.2.

As explained also in the manuscript and in our reply to other referees, we use an undetermined Berry phase to characterize the topological invariant, which is not a topological number (we emphasize that a topological invariant does not need to be a number but could be any property that does not change under small deformations that do not change topology), not a normal Berry phase or a Chern number calculated from the Berry curvature. This is because we are calculating the topological invariant when the surface" enclosing the nodes is effectively zero dimensional, while not one dimensional (Berry phase) or two dimensional (Chern number from the integration of the Berry curvature). Now we explain why in our system thesurface" enclosing the nodes is effectively zero dimensional and what an "undetermined Berry phase" really means in the reply to the next sentence the referee wrote.

1. The referee said "I am also skeptical about the following procedure "Here as we are in one spatial dimension effectively and the manifold enclosing the node is in zero residual spatial dimension, the calculation of the topological invariant is different from the Berry phase or Berry curvature for nodal line or Weyl semimetals". I don't understand what topological number the authors get out from this procedure."

This is indeed a different topological invariant that is not the Berry phase or the Berry curvature. This is not a topological number: as we explained in the manuscript and in the reply to other referees, topological invariants do not need to be an number and it could be a property that does not change under deformations that do not change topology. Here we are using the property if the two adjacent states on the two sides of the node of the same band are orthogonal or not as a topological invariant, and we call this the undetermined Berry phase: this is only a terminology here, which in fact refers to the fact that the two adjacent states on the two sides of the node of the same band are orthogonal to each other. This is different from the normal Berry phase or Berry curvature for other dimensions.

This property could be a topological invariant because it does not change under deformations that do not change topology of the bands and whether the two states are orthogonal or not depends on whether they are topologically nontrivial or not. In the trivial phase, they are not orthogonal. In this sense, this property is a good candidate for a topological invariant.

1. The referee said"I fear that by reducing their 4D problem to a 1D one, they miss sound topological interpretations."

This question is also related to the next one. As we have already emphasized in the manuscript and in the reply to other referees, we are not "reducing a 4D problem to a 1D one", on the contrary, our system is 1D effectively when calculating topological invariants. We are not trying to reduce it. The fact is that for all systems that we consider, including the 4D single case and nD+nD cases, we either need a protecting symmetry or we do not need a protecting symmetry but the nodes form a two dimensional surface. For the former case, i.e. when we need a protecting symmetry, topological invariants need to be computed locally from high symmetry points of the Brillouin zone (also related to the next sentence from the referee). As we require two symmetries, the system becomes effectively 1D. This is the property of a symmetry protected topological state and it is not that someone is performing a dimension reduction by hand. Thus we have not missed sound topological interpretations".

1. The referee wrote Usually, in band insulators, topological indices can be computed locally from high symmetry points of the Brillouin zone, where the sign of some relevant quantity is evaluated (see e.g. Z2 topological index in 3D time-reversal invariant topological insulators with inversion symmetry)."

It is not true that "Usually, in band insulators, topological indices can be computed locally from high symmetry points of the Brillouin zone" in general. High symmetric points are only required when the system requires a protecting symmetry. Of course, the example the referee raised indeed needs a protecting symmetry: the inversion symmetry.

In our cases when a protecting symmetry is required, we indeed calculate the topological invariants at high symmetric points, which makes the system 1D effectively. In other cases where protecting symmetry is not needed, the nodes form two dimensional surfaces in the momentum space and we still need to calculate the topological invariant in 1D effectively, as could be seen clearly from the Fig. 2 of arXiv. 1609.05414.

1. The referee wrote "Another example, maybe closer to what is discussed in that manuscript, would consist in looking at the Chern numbers associated to each band crossing in 3D parameter space instead of 1D parameter space, i.e. by fixing one of their 4 dimensions."

We are focusing on the topological invariants of each band crossing in the 3D parameter space. It is just that the calculation of symmetry protected topological invariants require that we perform the calculation in 1D effectively. Thus we emphasize again that we are not reducing the system to 1D by hand but the calculation of the topological invariant for symmetry protected states or for spherical nodes require us to work in 1D effectively.

1. The referee wrote "Another procedure to characterize band crossing points is by the homotopy properties of the map k -> h(k), which in Weyl nodes H=h(k).sigma , is encoded into the homotopy group of the sphere \pi_2(S^2)."

Indeed this is a way to characterize different equivalent classes of topology, i.e. how topology is classified, but this does not give topological invariants directly. Also this is better used when the enclosing surface is one or two dimensional, corresponding to \pi_1 (Hamiltonian manifold) and \pi_2 (Hamiltonian manifold) separately. Here we emphasize again that we do not have a two dimensional sphere enclosing the node when calculating the topological invariants as we are either working at high symmetric points like fig.a of fig 2 in arXiv.1609.05414 or we are having nodes forming a two dimensional surface so that the enclosing surfaces" could only be zero dimensional.

1. The referee wrote "I can understand that one of those procedures would yield a well-defined (even though maybe not sufficient) topological characterization of their band crossing points. But in the present form, I must say that I am not convinced by the topological characterization of those nodes. They probably are, but, in my opinion, this remains to be shown."

We would like to thank the referee for various suggestions to define a topological invariant in this case. As we have explained, the enclosing surface" is zero dimensional effectively, which makes some definitions of topological invariants not applicable, e.g. the Chern number, the Berry phase, etc.. Here we have defined our own topological invariant in the zero dimensional case, though it is not a number. In fact, the effectively zero dimensional case also exists in the three spatial dimensional nodal line systems when they require a protecting mirror symmetry. This could be found around eqn. (6) of arXiv. 1609.05414. We also would like to define a similar topological number like eqn. (6) of their case. However, there are several reasons that we did not proceed in this direction. First, in our case, the protecting symmetries are two special spacetime symmetries in the y and z directions, making it difficult to directly generalize their definition of the topological number. Second, even though we could define a topological number in this way, we have several systems in which the calculation of topological invariants are still at zero effective dimensions but there are no protecting symmetries, which also makes the generalization of that topological number difficult as that procedure requires a protecting symmetry. Finally, we believe that if there is an appropriate generalization of this topological number into our case, the result and mechanism would be the same as the one we defined, though we would also need to show that the generalized definition of that topological number is indeed a topological invariant.

Finally we would like to discuss more with the referee regarding the topological invariant, especially if a generalization of the topological number in arXiv.1609.05414 could be defined as well. However, we believe that our work is complete in that we have shown that our definition of a topological invariant is indeed a property that is not possessed by the trivial state while does not change under perturbations. We also need to emphasize that we have shown that the gapless system is robust under small perturbations where gaps would not open confirming that the system is topologically nontrivial even without the necessity of defining a topological invariant. Nevertheless we would be happy to see if there is another possible definition of a topological invariant which is a number in this case, possibly along the lines of eqn. (6) in arXiv. 1609.05414.

Author Ya Wen Sun on 2021-05-09
(in reply to Report 3 on 2021-05-05)
Category:
validation or rederivation

This is a second reply to the third referee as well as other two referees. First we would like to thank all the referees for helpful questions. Being stimulated by the questions from the referees, we started to rethink about the possibility to define topological invariant which is a topological number like many other topological systems. We follow the line of equation (6) of arXiv. 1609.05414 and we find that indeed we could define a similar topological number in our case, too. In our case, the protecting symmetry would be a combination of reflection symmetry in both y and z directions and depending on the eigenvalues of this symmetry we could distinguish the two bands by their eigenvalues. Then we could find that indeed we have a nontrivial topological number using equation (6) of arXiv. 1609.05414. Note that this is based on all our previous replies where we explained why we need to calculate the topological invariant in an effective zero dimensional system. We will revise our manuscript recently and add this calculation as well as explanations about the protecting symmetry and the effective zero dimension to the manuscript. We will also show why this definition of the topological invariant (as well as the case in arXiv. 1609.05414) is the same as our previous definition of the topological invariant in our revised manuscript. We would be happy to answer more questions from the referees if there are still confusions from all referees.

Report

I thank the authors for their detailed responses to my comments and the changes to the paper, as well as for their response to the other referee (which I have also benefitted from). The changes have clarified various aspects which I found confusing (notably, I understand better the role of ky = 0). Unfortunately I still remain unconvinced of the fundamental existence of a topological character. A few more detailed points on this:
-- I remain somewhat perplexed by the non-existence of a topological invariant; though mention is made that the orthogonality of various states should be viewed as being equivalent to the existence of an invariant, I am not convinced of this. Indeed in this new version the authors emphasize (on p12) that the gapless point in momentum space is indeed a *point* in momentum space; this is particularly important in relation to my comments on the role of ky). But if it is a point should there not then exist an integral around this point that measures the appropriate Berry phase? (I also note that the Nature paper https://www.nature.com/articles/s41567-019-0561-1 cited by the authors in their most recent comment does exhibit a non-zero topological invariant).
-- A further reason for my skepticism arises from the fact that the physics described can occur when considering hydrodynamics in a different coordinate system (but still on flat space). I brought this up in a previous comment, as I do not feel that a fundamental property such as whether or not a mode is topologically protected should depend on the coordinate system used to describe the physics. In their response the authors drew a distinction between the notions of coordinate system and reference frame that I confess I could not fully understand. I believe a simple physical explanation as to why a topologically protected gap should appear in a different coordinate system is not yet transparent in the paper.

I must thus regretfully say that the author's statements have not fundamentally changed my view on the paper. I believe a computation of a non-trivial topological invariant would however be a convincing demonstration of the topological character of the mode; in the absence of this however I cannot recommend publication in its current form.

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Author Ya Wen Sun on 2021-04-26
(in reply to Report 2 on 2021-04-25)

First we would like to thank the referee for reading our revised manuscript! The referee raised again two questions and we will answer them one by one.

1 Regarding the referee's first question about the topological invariant, there is something that we have explained in the text related to the dimensionality of the system and there is something else that we need to clarify here regarding the difference between gapped topological states and gapless topological states. We will explain both of these two pints below.

1) Topological invariant regarding the dimensionality of the system.

For a normal Weyl semimetal in 3+1 dimensions, the momentum space is three dimensional (kx, ky, kz dimensions). The node is a point in this three dimensional space. To calculate the topological invariant for that case, we need to integrate the Berry curvature on a two dimensional sphere enclosing this point. This is special for calculating the topological invariant on an effectively two dimensional (the sphere enclosing the node) space.

Then for nodal line semimetals in 3+1 dimensions, the nodes form a circle in the three dimensional momentum space. To calculate the topological invariant in this case, we only have a circle enclosing each nodal point so the calculation of the topologicla invariant is on an effective one dimensional circle. In this case we do not integrate the Berry curvature but instead directly calculate the Berry phase accumulated on this circle.

This is the difference due to dimensionality. An illustration picture is shown clearly in Fig. 2 of arXiv. 1609.05414. (Sorry we cannot append a picture in this reply form, which we intended to append even in our last reply, so we have to kindly ask the referee to check this paper for this figure. Thanks!) In Fig.2, the Weyl semimetal case could be the circle shrinking to point limit of c) in that figure. The nodal line is the second b) case. Our case as emphasized corresponds to the effectively zero dimensional case, i.e. case a) in Fig.2.

We have effectively zero dimensional because in some cases we have a sphere of nodes so that effectively the dimensionality to calculate the topological invariant is zero dimensional or in some cases the system is symmetry protected by translational symmetry in the y and z directions so that we have calculate the topological invariant in a high symmetric point which also makes it effectively zero dimensional. Thus the calculation of our topological invariant is different from the effectively two dimensional Berry curvature and the effectively one dimensional accumulated Berry phase.

Here the same as Fig.a) in Fig. 2 or arXiv. 1609.05414, the topological invariant could only be to compare whether the two eigenstates at the two points on the same band are orthogonal or not. As long as this property does not change under small perturbations, it could serve as a topological invariant. This is the definition of a topological invariant. And as long as the quantity is different from its corresponding value for a trivial state, it is called a topologically nontrivial invariant.

We have checked that with different m, b, as long as it is in the topological nontrivial phase, the two points on the same band are orthogonal and this is different from a trivial band where the two eigenstates are not orthogonal but are the same up to a relative arbitrary phase.

This is the first point of why our calculation of the orthogonality could serve as a topological invariant and why it is different from the Weyl semimetal's Berry curvature or nodal line semimetal's Berry phase calculation.

2)Regarding the difference for topological invariants between gapped topological states and gapless topological states.

In order not to confuse the referee by giving too many details in our last reply to the other referee, we did not mention that the nature physics paper, which uses the same effective Hamiltonian as ours, focuses on the gapped topological states while we focus on gapless topological states. Famous examples of gapped topological states are topological insulators. To calculate the topological invariant of a topological insulator, they require integration in the whole Brillouin zone but for gapless topological states of matter, there is one special node in the momentum space so to calculate the topological invariant the system is effectively one dimensional lower and we only integrate on a co-dimension 1 surface enclosing this node. Thus we have effectively zero dimensionality here to calculate the topological invariant and no integration is needed in comparison to the case of gappled topological states.

Second, why do we say their paper is a piece of proof that our gapless topological states of the same system is indeed topological? This is because gapless and gapped topological states usually have a special relationship. For example, Weyl semimetals usually appear near the critical point when a topological insulator has a phase transition to a normal insulator. For a gapped topological state to become a normal gapped state through a phase transition, it has first to pass a gapless state, i.e. its gap has to close first. This is the reason that it is called topologically nontrivial gapped states as normal gapped states could deform to trivial gapped states without the need of a gap closing. Thus usually when a system has a nontrivial gapped state, topologically nontrivial gapless states would appear at the gap closing parameters and Weyl semimetals could appear during the phase transition between some kinds of topological insulators and normal insulators.

2 Regarding the second question of why reference frame transformations would change the spectrum, also why would it change whether the system is topological. We explain this in several aspects.

1) This question could also be raised for the famous Unruh effect. The Unruh effect states that a vacuum state observed by an inertial observer while observed by an accelerating observer would look like a finite temperature thermal state. Thus whether it is a vacuum state, the temperature, the property of whether the system is a thermal state or not, all depend on observers, i.e. reference frames. It is no surprise when now we get to know that whether a system is topologically nontrivial or trivial would also depend on observers, or equivalently reference frames. We will also give more details on why a system is topologically nontrivial or not would depend on reference frames in the next points.

2) Here in our case, the basic ingredient is that the spectrum would change under reference frame change. The Unruh effect is also an example that the spectrum of a system would change under reference frame change. An even simpler case would be the change of the spectrum under a boost transformation, i.e. for inertial observers with a nonzero velocity compared to the current reference frame. In that case, it could be checked that the normal spectrum would be rotated for a small angle (related to the boost parameter) on the E-k plane. The structure of the spectrum would be even more complicated for non-inertial observers and the spectrum would be quite different when being observed by accelerating observers.

3) We have another more detailed technical explanation on why the spectrum would change from the viewpoint of the change of frequencies and momenta under reference frame change. This will be presented in our new work to appear in one or two months. We describe it here briefly. Under a reference frame change, the frequency and momentum would be transformed accordingly, which makes the spectrum change. In our case, it looks strange that in the original frame, there is only crossing point but in the new reference frame, there are four crossing nodes. An immediate question is how one crossing point changes to four crossing nodes. This is really something that should not change due to reference frames. The fact here is that: under our reference frame change, real spectrum would become complex while orginal complex spectrum would become real! Thus here the one crossing node in the orginal frame becomes complex which cannot be observed in the new frame, and the four nodes observed in the new frame come from four crossing nodes in the original complex spectrum space which cannot be observed in the original frame. This explains why a topologically trivial system could become nontrvial: the now topologically nontrvial bands were in fact hided in the complex spectrum and cannot be observed in the original system. And the previous topologically trivial system is now hided in the complex spectrum and cannot be observed in the new accelerating system.

4) Finally we explain the distinction between a coordinate transfermation and a reference frame change. In circumstances that do not cause confusion, we could use them interchangingly, but of course, in fact they are different physically. For example, the polar coordinate system and the Cartesian coordinate system are different coordinate systems so we need to match their observables and they should observe the same physics. However, if we fix the Cartesian coordinate system, and then perform a coordinate transformation and in the new coordinate system the observer is again in a Cartesian coordinate system, then this is in fact a reference frame transformation. Here in our work, we are in fact talking about reference frame transformations so that two systems could observe very different physics.

We hope these clarify the referee's confusion and we believe our work is solid enough to be published. We would be very happy to discuss with the referee on further questions that he/she would raise, which we would also benifit from. Thanks!

Finally, we would need to emphasize that our current paper is a generalization of our previous 2004 paper, which has already been published in PRD after peer review by two experts in the field (Phys.Rev.D 103 (2021) 4, 044044). All the questions the two referees raised in these reports are about that paper while not about our new work in this paper. We believe our current work should be solid enough to be published in a high quality journal.

See report.

Report

I have read the revised version of the manuscript and the authors' extensive replies to my comments, and to the question and comments of the second referee who seems to have had several concerns that went along the same lines as my own concerns. While I appreciate the thoroughness of the authors in replying and amending the text, my opinion of this paper has unfortunately not changed. I still believe that this is an ad hoc modification of hydrodynamics that produces certain (engineered) results that are then by analogy claimed to be those of a topologically protected state. No part of the paper convinces me that this goes beyond what is called 'argument from analogy'. From the replies of the authors, this topological nature moreover seems to be somewhat of an axiom rather than something that has been convincingly established. For this reason, it is my opinion that the paper should be rejected by SciPost.

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Author Ya Wen Sun on 2021-04-27
(in reply to Report 1 on 2021-04-22)
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As both referees are not familiar with topological states of matter, let me add more basic knowledge on topological states of matter here. This could be viewed as a supplementary reply to both referees.

1 There are gapless and gapped topological states of matter, which are different in several aspects. Gapped topological states cannot be adiabatically deformed to a normal gapped state without closing the gap in the process. In contrast, a gapless topological state cannot become gapped by very small perturbations of the system ( under required symmetries).

Thus to confirm if a gapless state of topologically nontrivial or not, we need to know if the band crossing (i.e. the crossing node) is an accidental crossing or not. If a small perturbation could gap it, then it is an accidental crossing and if small perturbations only change the position of the crossing nodes, it is a topologically nontrivial gapless state.

This is why we could tell if the gapless system is topologically nontrivial even before we perform a topological invariant calculation. We could already know if it is topologically nontrivial by studying the spectrum of the system under deformations of the parameters of the system. Of course, in our work we also calculated the topological invariants as further confirmation.

2 In our work, we performed the topological invariant calculation as a further confirmation that our system is topologically nontrivial. As could be found in the standard textbook, a topological invariant is by definition a property that remains the same as long as the topology does not change. A topological invariant does not need to be a number. In our work, we take the property of whether the two states on the two sides of the node in the same band are orthogonal or not to be a topological invariant.

Then we first showed that no matter how parameters of the system change (e.g. m, b parameters), as long as the system is in the topologically non-trivial phase, the orthogonal property does not change, i.e. it is by definition a topological invariant that describes a nontrivial state. The trivial state (the one that could be deformed to a vacuum state without passing through any singular point) has the property that the two states are the same up to a relative phase, so the two states cannot be orthogonal in the trivial phase, which we also confirmed.

This is why we call this orthogonal property a topological invariant. As we explained in the last reply, this is also a zero-dimensional analog of the two-dimensional Berry curvature integration and the one-dimensional Berry phase integration for topological invariants calculation of gapless states. Note that this is different from the calculation of topological invariants for gapped states as we explained in detail in my reply to the second referee.

Finally, we would be happy to explain more whenever the referees have any confusion about topological states of matter.