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:

Analog Systems for Gravity Duals
S. Hossenfelder
Phys. Rev. D 91, 124064 (2015)
arXiv:1412.4220 [gr-qc]

and I have slides of my talk here.