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Topological hydrodynamic modes and holography
by Yan Liu, YaWen Sun
This is not the current version.
Submission summary
As Contributors:  Ya Wen Sun 
Preprint link:  scipost_202011_00004v1 
Date submitted:  20201105 06:41 
Submitted by:  Sun, Ya Wen 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

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 nonconservation 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:
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021121 (Invited Report)
Strengths
The paper is very clearly written and describes an intrinsically interesting topic. It attempts to weave together different strands of physics (topologically protected gaplessness and hydrodynamics) in a novel manner. Furthermore, hydrodynamic predictions are tested against holographic constructions in a robust way.
Weaknesses
Unfortunately, I feel the central point of the paper  that, through deformations of the hydrodynamic conservation equations, it is possible to obtain topologically protected excitations in hydrodynamics  has been insufficiently justified in the paper. I discuss below what I see as the current shortcomings and how they might be improved.
Report
In this paper (which is an extended version of Ref [6]), a detailed description of novel hydrodynamic phenomena is given, where it is argued that certain deformations of the usual hydrodynamic equations (obtained from a controlled breaking of momentum conservation) give rise to collective modes that have a topological character, making analogies to wellknown examples such as Weyl semimetals. The possible deformations are described in detail, and it is explained how they can arise from various "microscopic" considerations (e.g. considering hydro on a nontrivial gravitational background, or coupling to an external field). The paper is wellwritten and interesting; however I feel that the main interesting point regarding the topological character is somewhat insufficiently justified.
To summarize: e.g. around Figure 3 on p10 it is explained that there are two phases of the system as a function of parameters called (b,m); in particular, when b > m there are two "band crossings", which vanish when b = m. It is argued that these band crossings are an example of a topologically protected mode. I am afraid I remain unconvinced of this basic point. My issues are:
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)
2. It remains a bit unclear what exactly is topologically protected.
The second point is addressed in section 5, where a potential topological invariant is discussed. Perhaps I am confused by the lowdimensionality, 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.
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).
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.
I do stress that despite my criticism above, I find the results interesting and feel that the paper could benefit from further explanation of the confusing points above. If the emphasis on the topological character is to be retained, I feel they need to be explained further. However if the paper is to be published simply as an exploration of deformations of hydro it needs less revision, but still I feel the points above deserve further discussion.
Requested changes
1. Further explanation of the topological invariant.
2. Further explanation of the nature of the symmetry that protects the dispersion relation.
3. Further discussion on why the existence of bandcrossing points should be viewed as analogous to \omega = 0 in more traditional examples of physics.
Anonymous Report 1 on 20201221 (Invited Report)
Report
The paper on Topological hydrodynamic modes and holography by Liu and Sun is an exploration of hydrodynamic modes in systems without energymomentum conservation. The paper is a long version of 2004.13380 written by the same authors.
The main claim of the two papers seems to be that by modifying the energymomentum conservation Ward identity in a specific way, one obtains gapped hydrodynamic modes, which must be related to nontrivial topology. While this is certainly an interesting claim, at present, I am unconvinced by the submission that this makes sense. While the first part of the claim is certainly correct (modified Ward identity can lead to gapped modes), I do not see how the authors reach the conclusion that this is somehow connected to topology. As a result of this, at present, I cannot recommend that this paper be published by SciPost.
Let me explain this conclusion in somewhat greater detail. Hydrodynamics with conserved energy and momentum gives rise to gapless diffusive and sound modes. When the energymomentum Ward identity is modified (e.g. by momentum nonconservation discussed in Appendix A), it can lead to gapped dispersion relations. This is nothing special. It happens for the example in Appendix A, where the authors seem to reach the conclusion that this is somehow different to their studied examples because there is a nontrivial contribution to the imaginary and not the real part of the spectrum. 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"?
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. Why does this have to imply (or rather, be a consequence of) nontrivial topology? I do not think that this is explained in this work. 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. 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)?
I suspect that the answer is no. One reason are precisely the holographic examples that the authors show, which do not seem to have any relation to topology, but are very closely related to the kind of considerations made by Ref. [30] (and numerous subsequent works) that led precisely to what the authors describe in Appendix A.
Author: Ya Wen Sun on 20201222
(in reply to Report 1 on 20201221)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 energymomentum conservation equation leads to nontrivial topological gapless states. This specific modification could come from a change of reference frame from an inertial one to a noninertial 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) nontrivial topology? I do not think that this is explained in this work.
Our answer:
This is not a consequence of nontrivial topology, so the first part is the correct question, i.e. this implies nontrivial topology. Let us answer why this implies nontrivial 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 nontrivial 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.