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Topological hydrodynamic modes and holography

by Yan Liu, Ya-Wen Sun

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
  • Gravitation, Cosmology and Astroparticle Physics
Approach: Theoretical


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.

Our answer:

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.

Our answer:

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, 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.

Our answer:

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.

Our answer:

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, 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.

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Resubmission scipost_202011_00004v2 on 17 February 2021

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