Thursday, July 02, 2015

Analog Duality

I have recently begun working on a research topic entirely new for me. The idea is to combine analog gravity with the applications of the gauge-gravity duality to condensed matter systems. Two ingredients are necessary for this: First, there is the AdS/CFT duality that identifies a gravitational system in asymptotic anti-de Sitter (AdS) space-time with a conformal field theory on the boundary of that space. And second, there is analog gravity, which is the property of some weakly coupled condensed matter systems to give rise to an effective metric. Combine both, and you obtain a new relation, “analog duality,” that connects a strongly coupled with a weakly coupled condensed matter theory, where the metric in AdS space plays the role of an intermediary.

I have now given several seminars about this, and the most common reaction I get is “Why hasn’t anybody done this before?” Well, I don’t know! It might sound obvious, but it isn’t all that trivial once you look at the details. For this junction of two different research directions to work, you have to show that it is indeed possible to obtain the metrics used in the gauge-gravity applications as effective metrics in analog gravity.
The gauge-gravity duality (left) identifies some solutions in General Relativity with strongly coupled condensed matter systems. Analog gravity (right) identifies some solutions in General Relativity with effective metrics of certain types of weakly coupled condensed matter systems.

Provided one can show that the set of metrics used in the gauge-gravity duality overlaps with the set of metrics used in analog gravity, one obtains a new relation, “analog duality,” between strongly and a weakly coupled condensed matter systems.

Concretely, for the applications of the gauge-gravity duality to condensed matter systems people do not use the full mathematical apparatus of this correspondence. To begin with they only work in the limit where in the AdS space one has only classical gravity, and some fields propagating in it, but no string effects. And they are not using a system that is known to correspond to any well-understood theory on the boundary. These are phenomenological models, which is why I find this research area so appealing. Mathematical beauty is nice, but for me describing reality scores higher. For the phenomenological approach one takes the general idea to use a gravitational theory to describe a condensed matter system, and rather than trying to derive this from first principles, sees whether it fits with observations.

In my paper, what I have done is to look at the metrics that are being used to describe high temperature superconductors, and have shown that they can indeed be obtained as effective metrics in certain analog gravity systems. To prove this, one first has to convert the metric into a specific form, which amounts to a certain gauge condition, and then extract the degrees of freedom of the fluid. After this, one has to show that these degrees of freedom fulfil the equations of motion, which will not in general be the case even if you can bring a metric into the necessary form.

Amazingly enough, this turned out to work very well, better indeed than I expected. The coordinate systems commonly used in the AdS/CFT duality have the metric already in almost the right form, and all one has to do is to apply a little shear to it. If one then extracts the degrees of freedom of the fluid, they quite mysteriously fulfil the equations of motion automatically. The challenge in that wasn’t so much doing the calculation but to find out how to do it in the first place.

I found this connection really surprising. The metrics are derived within classical General Relativity. They are solutions to a set of equations that know absolutely nothing about hydrodynamics; it’s basically Maxwell’s equations in a curved background with a cosmological constant added. But once you have learned to look at this metric in the right way, you find that it contains the degrees of freedom of a fluid! And it is not the fluid on the boundary of the space-time that belongs to the gauge-theory, but a different one. This adds further evidence to the connection between gravity and hydrodynamics that has been mounting for decades.

One then has two condensed matter systems describing aspects of the same gravitational theory, which gives rise to relations between the two condensed matter systems. It turns out for example that the temperature of the one system (on the boundary of AdS) is related to the speed of sound of the other system. There must be more relations like this, but I haven’t yet had the time to look into this in more detail.

As always, there is some fineprint that I must mention. It turns out for example that both the analog gravity system and the boundary of the AdS space must have dimension 3+1. This isn’t something I got to pick; the identification only works in this case. This means that the analog gravity system describes a slice of AdS space, but unlike the boundary it’s a slice perpendicular to the horizon. The more relevant limitation of what I have so far is that in general the equations of motion for the background of the fluid in the analog gravity system will not be identical to Einstein’s field equations. This means that at least for now I cannot include backreaction, and the identification only works on the level of quantum field theory in a fixed background.

I have also in the present paper worked only in a non-relativistic limit, which is a consistent approximation and fine to use, but if the new relation is to make sense physically there must be a fully relativistic treatment. The relativistic extension is what I am working on right now, and while the calculation isn’t finished yet, I can vaguely say it looks good.

So I’m still several steps away from actually proving that a new duality exists, but I have shown that several necessary conditions required for its existence are fulfilled, which is promising. Now suppose I’d manage to prove in generality that there is a new duality that can be obtained by combining the gauge-gravity duality with analog gravity, what would it be good for?

To begin with, I find this interesting for purely theoretical reasons. But beyond this, the advantage of this duality over the gauge-gravity duality is that both systems (the strongly and the weakly coupled condensed matter system) can be realized in the laboratory. (At least in principle. I have a Lagrangian, but not sure yet exactly what it describes.) This means one could experimentally test the validity of this relation by comparing measurements on both systems. Since the gauge-gravity applications to condensed matter systems were indirectly used to obtain the new duality, this would then serve as an implicit experimental test for the validity of the AdS/CFT duality in these applications.

A year ago, I knew very little about any of the physics involved in this project. I’ve never before worked on AdS/CFT, analog gravity, magnetohydrodynamics, or superconductivity, and this is all still very new for me. It’s been a steep learning curve, and I am nowhere near having the same overview on this research than I have on quantum gravity phenomenology – there I’ve reached a level where it feels like I’ve seen it all before. I’m not good with planning ahead my research because I tend to go where my interest takes me, but I think I’ll stick around with this topic and see how far I can push this idea of a new duality. I definitely want to see if I can make the case with backreaction work, at least in the simplest scenarios.

For more details, the paper is here:
and I have slides of my talk here.


  1. Analog models don't have any holagraphic entropic bound, or even any entropic bound, for that matter. I think to get AdS/CFT, you'd need at least to have a holographic entropy bound.

  2. CF,

    Well, first this depends on the particular system you look at. That there is no known bound in the systems used so far doesn't mean no system has one. Besides this, I don't know how the entropy of the analog system relates to the entropy of the gravitational system (or its dual).

  3. I think the holographic entropic bound shows up via the Ryu-Takayanagi formula.

    Analog models have ordinary quantum field theory as a substrate, and so, they can't have any entropic bounds, at least within their effective field theory range of validity.

  4. CF,

    I don't know how in general the degrees of freedom of the background correspond to those of the metric, and whether all possible configurations of the fluid have a corresponding metric in AdS, I suspect not (because there will still be some boundary condition). So the answer is, I don't know, but thanks for bringing it up.

  5. It's wonderful. You have the right approach. I'm so happy. Hope you'll get the solution soon and I'm sure you'll get it.

  6. "rather than trying to derive this from first principles, see whether it fits with observations" GR is geometric. Test gravitation geometrically - an orthogonal observation then effecting a path correction. "Mathematical beauty" need not be empirical. Answers are where they are, not where they should be.

    "high temperature superconductors" Hot times!
    arXiv: 1402.2721, 1412.0460, 1501.01784, 1502.02832
    Phys. Rev. Lett. 114 157004 (2015), doi: 10.1103/PhysRevLett.114.157004

    "connection between gravity and hydrodynamics" All matter is fermionic.
    Phys. Lett. B 697(4) 404 (2011), doi:10.1016/j.physletb.2011.02.041
    arXiv:1210.2186, 1203.4259

    Bee wonderful.

  7. Using dualities to analyze strongly coupled quantum field theories from condensed matter physics (and elsewhere) seems like one of the most exciting research topics these days. Good luck! I look forward to reading blog posts about what you discover.

  8. A fluid, that's interesting, it changes of the pure jelly of GR. Is it compatible with a sea of heavy bosons produced by the energy of the field? If it's a fluid it's surely a Lagrangian-Poisson; Poisson is fish in French :-)

    Your productivity is beyond me, I can barely simply follow the blog, your nickname Bee is more than well deserved.

  9. "Mathematical beauty is nice, but for me describing reality scores higher. For the phenomenological approach one takes the general idea to use a gravitational theory to describe a condensed matter system, and rather than trying to derive this from first principles, sees whether it fits with observations."

    I can understand your enthusiasm because mine goes exactly in the opposite direction. I would thus write your sentence as: "Describing (factual) reality is nice, but for me mathematical beauty scores higher. [...] Rather than trying to see whether it fits with observations, one derives it from first principles."

    I am usually excited when I see these two words "first principles" and it is then when I am encouraged to pay attention. I find observed factual reality not so interesting. The interesting thing is what lies at the mysterious and always unreachable heart of things. These are the First Principles. Mathematical first principles are a beautiful toy that accompany the higher First Principles. Something that one measures with an apparatus is, definitely, not so interesting and I find fitting curves a boring exercise. One does that often to exploit the things in the world and to introduce new engines in the world. These are second or third principles (sometimes total lack of principles) but not first principles. (I would say.)

    As the poet Keats said "Heard melodies are sweet but unheard melodies are sweeter".

  10. Sabine - this got anything to do with Lee Smolin per chance? Or could it be he's a major influence.

    It's just that the solution architecture you bring into play, is characteristically very much in the vein of 'trialities' - his recent paper showing the duality as framed now, is inadequate for effectively leaving implicit the intersection work, and what applied treatments accomplishing that entailed.

    It's possible I'm mixing things up here...I probably haven't absorbed either proposition effectively. That said I think I'm right.

  11. Chris,

    Not sure why you're asking this. No, of course it's got nothing to do with Lee Smolin who, for all I know, has never worked neither on analog gravity nor on applications of AdS/CFT to condensed matter system, or on magneto-hydro, or anything else related to this paper. Or if he has, I haven't read it.

  12. yeah sorry about that. For some reason my articulation came off laboured, which basically looks a lot like when something negative is intended in some deniable way (the commonality is the labouring).
    But I was just labouring....probably because about half way through I began to have doubts about the analogue I was making, which you know...causes rewriting and re-phrasing, trying to cover one's arse with suitable vagueness so to manage the doubts.
    But it was totally innocent. All I meant to say was "have you seen Smolins's paper on trialities" or is he an influence. Only because I know you've been in touch recently.
    I can see that the wording could be interpreted as a possible allegation, or insinuating a dirty secret or whatever. That was totally not there at the time of writing!Rgds.

  13. Chris,

    Apology accepted. I've briefly looked at that paper, yes. It came out 3 months after mine, so if it was an influence it must have been acausal ;)

  14. Reading your very interesting post is the first time I've come across the phrase "strange metals" in a scientific context. I wasn't quite sure what it meant, so I googled the phrase, and there was quite a bit of literature about it. The one article, easiest for me to understand, as it was written for a lay audience, was by Subir Sachdev, of Harvard University in the January, 2013, Scientific American (oops, just noticed that you referenced Subir Sachdev in your post).

    This is all very fascinating, and hopefully you can develop more quantitative predictions for these high temperature superconductors that are accessible to experimental verification.

  15. I've been reading 'The Maxwellians' about Heaviside, Lodge, Fitzgerald et al and how they turned Maxwell's electromagnetic theory into the classical EM theory we know today. (Heaviside was the guy who formulated Maxwell's equations.) One interesting thing was their use of mechanical models, often built out of pulley wheels, elastic bands and so on that were used to explore the electromagnetic field. The equations describing these models were similar to those that described the field, so they could often tell when their intuition was on or off track. For example, analog models of electrical conduction involved energy stored in the stretching of the various elastic bands, not along the "conducting" slot. This confirmed their growing sense that the energy in a current was not passing through the wire, but through the external field. (Basically that Poynting was right.)

    Analog gravity sounds like a similar approach. As the Maxwellians would have put it: the idea is to build an analog for quantum gravitational system, not a likeness. Needless to say, as the 19th century waned it became increasingly obvious that electromagnetic waves weren't at all like mechanical systems built with pulley wheels and rubber bands, but building analog models seems like a good way to improve our understanding.


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