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Topological hydrodynamic modes and holography
by Yan Liu, Ya-Wen Sun
This is not the current version.
|As Contributors:||Ya Wen Sun|
|Date submitted:||2020-11-05 06:41|
|Submitted by:||Sun, Ya Wen|
|Submitted to:||SciPost Physics|
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.
Submission & Refereeing History
- Report 3 submitted on 2021-05-05 14:22 by Anonymous
- Report 2 submitted on 2021-04-25 23:32 by Anonymous
- Report 1 submitted on 2021-04-22 18:19 by Anonymous
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Reports on this Submission
Anonymous Report 2 on 2021-1-21 (Invited Report)
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.
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.
In this paper (which is an extended version of Ref ), 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 well-known examples such as Weyl semi-metals. 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 non-trivial gravitational background, or coupling to an external field). The paper is well-written 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 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.
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.
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 band-crossing points should be viewed as analogous to \omega = 0 in more traditional examples of physics.
Anonymous Report 1 on 2020-12-21 (Invited Report)
The paper on Topological hydrodynamic modes and holography by Liu and Sun is an exploration of hydrodynamic modes in systems without energy-momentum 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 energy-momentum conservation Ward identity in a specific way, one obtains gapped hydrodynamic modes, which must be related to non-trivial 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 energy-momentum Ward identity is modified (e.g. by momentum non-conservation 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 non-trivial 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) non-trivial 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.  (and numerous subsequent works) that led precisely to what the authors describe in Appendix A.