123. AdS/CFT and condensed matter applications
COND-MAT, HEP-TH/PH — By Dmitry Podolsky on December 4, 2008 at 10:00 am
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This post is going to be, I think, somewhat controversial 
but… if you feel that I greatly miss some important point regarding the subject, then please feel free to explain that to me in the comments. And the subject is… ta-da-da-daam…
how exactly string theory may help us to solve some problems in condensed matter physics!
There is currently huge body of papers that exploit the idea of AdS/CFT duality to approach well known problems in condensed matter physics such as high temperature superconductivity, quantum Hall effect, Nernst effect etc.
The KITP/Princeton group (Sean Hartnoll, Steven Gubser, Gary Horowitz, Chris Herzog) demonstrates, I think, the highest activity in this respect, but there are also other very important people such as Subir Sachdev who are strongly involved, so at some point I have felt that I need to understand what is about, otherwise I am in suburbia of contemporary theoretical physics
In particular, I am going to focus on superconductivity. What is the buzz? There exist materials (such as cuprates) called unconventional superconductors. Often these materials are also high temperature superconductors (current record material, I think, is some cuprate perovskite with 
K, the temperature corresponding to the normal pressure; if you increase the pressure, 
can be also somewhat increased but not very much). These superconductors are called unconventional, because presumably they are not described by the Bardeen-Cooper-Schrieffer theory: it looks like the mechanism of coupling for Cooper pairs is different than the one BCS proposes (in BSC, phonons are responsible for the coupling). However, some of high temperature superconductors (such as 
) are described by BCS, but the electron-phonon coupling seems to be strong.
If a theoretical physicist finds that some theory is in the regime of strong coupling, what does she do? Right, she starts looking for a dual description of the theory, and we all know how successful AdS/CFT was to explain strong coupling physics of 
super Yang-Mills.
Why not try using AdS/CFT here, you say? One can take a 3+1 dimensional gravity on Anti de Sitter background (this corresponds to a 2+1 dimensional CFT in the dual description), put a black hole in this setup (this would model a finite temperature phase of our CFT) and vary the size of its horizon to test the phase diagram of the CFT.
There are three serious problems with this way of thinking, as far as I can see.
1) AdS/CFT describes duality between gravity and a super Yang-Mills, that is, gauge theory. Behaviour of the strongly coupled gauge theory can be described in terms of weakly coupled gravitational degrees of freedom in higher number of dimensions.
But how does the gauge theory have anything to do with superconductivity??? All those high materials at best exhibit some global symmetries, not local (that is, gauge) ones. A gauge symmetry very strongly constrains dynamics of the theory, and would such a symmetry exist in high superconductors, it would be immediately seen in the correlation functions of observables (I mean, come on, observables themselves would correspond to gauge invariant operators )
It seems that the authors themselves are well aware of this issue. As Steven Gubser says in his recent paper,
Any such analog with high
not skepticism, because the underlying degrees of freedom of duals tovacua are typically large
gauge theories, seemingly distant from semi-realistic constructions
like the Hubbard model.
Well, they are even more distant from a somewhat more realistic BSC at strong coupling.
2) AdS/CFT is the duality between gravity and supersymmetric gauge theory. Even if you break SUSY by temperature corrections, the vacuum (that is, 
state) is still supersymmetric. As it seems, there are no signs of (even hidden) supersymmetry seen in high temperature superconductors. To be constructive, let me add that some disordered systems do admit supersymmetric behaviour (although SUSY is hidden quite well; the key name on the field here is Efetov), but it is unclear to me where disorder is present in the KITP/Princeton constructions.
3) AdS/CFT is the duality between gravity and conformal supersymmetric gauge theory. All these materials are maybe described by conformal field theory, but only on a single line of the phase diagram – at 
. If you take any other point, you will immediately see, say, the running of electron-phonon interaction coupling constant.
As a conclusion: my analysis is naive, and these guys are smart. Still, I have an impression that they play with AdS/CFT rather than solve a problem of physical interest. And to be completely honest, it is even unclear to me what exactly are they playing with.
6 Comments
Hi Dmitry! I hope you won’t forget to tell us the right answer to your latest poll.
Here.
1. Indeed, gauge theory looks different from the CMT systems. But I don’t think that your argument is enough to prove it. Gauge symmetry is not an inherent part of a theory: it is just a way to formulate certain theories. If condensed matter systems have no gauge symmetry to start with, it doesn’t mean that their physics in a limit cannot be fully equivalent to a gauge theory.
There is no physical way to “measure” how many gauge symmetries there are, exactly because they’re unphysical, and dualities relate theories/descriptions with different gauge symmetries.
2. All AdS/ion and AdS/CMT dualities I’ve seen are based on the expectation that physics in the limit is dominated by the gauge field, so whether or not you take the full supersymmetric theory or just its pure gauge theory segment is qualitativly irrelevant. This assumption raises concerns about the possibility to extend this duality to a fully exact description of things, but it makes it plausible that things can agree extremely well and at least give you a very useful intuition for physics.
3. Well, right. Non-conformal effects are either due to some spontaneous breaking or because you consider non-AdS/non-CFT from the scratch.
Hi Lubos
Currently, I am not so sure that my original answer to the poll was the correct one
, so I am still trying to figure out what happens. So far, the impression is the following.
Here is the quote from your previous comment, that made me think I might be wrong.
I think, this argument does not work – take a relativistic particle of mass m, not a string, and quantize it by first quantization. Then, there is a reparametrization invariance on the world line, which is exact. Still there perfectly exists a non-relativistic limit, when m is large enough. You just need to prepare an initial state |p> with p<>L, it should look like a particle).
But to get a spectrum of the BO Hamiltonian, we need internal excitations of the string, not the motion of the center-of-mass. Non-relativistic limit is achieved when v_\perp << c, and this should be also satisfied for low energy excitations of very long strings.
Yep, I already invented a counterexample myself: 3D Ising model is Kramers-Wannier equivalent to Z_2 gauge theory. This was the issue that bothered me most, so I am a kind of satisfied now.
Well, to see why my approximation works extremely well, I need to understand the area of its applicability. This, as you said, “raises concern”, since I am not quite sure what happens with duality if I break SUSY (probably clear if I break it from N=4 to N=2 not so clear if I break it further from N=2 to N=1, completely unclear, what happens if I break the last remaining SUSY). But I am not complaining – that’s how progress in any direction in physics works.
Cheers
Dear Dmitry, of course, I am troubled by the same thing concerning AdS/QCD and AdS/CMT – that the “rest of SUSY physics” is being neglected, and what it means, etc.
But maybe, the right way to treat is as a “new theory of real phenomena”. We should probably simply forget that we know that exact SUSY doesn’t exist in Nature.
We should just try to study the SYM description of the CMT systems and see how far we get. Of course, we won’t get to the very end. But it was normal in the history of physics that people had an approximate theory but they didn’t quite know how far it works.
Concerning the relativistic strings, your story would sound OK if you actually got the same answer in the small-velocity limit of the relativistic answer. But I don’t think you do.
In the case of particles, if you subtract the latent energy etc., the low-velocity limit of relativity is simply non-relativistic mechanics.
But if you start with relativistic fluxtubes, assuming it’s the right thing, you won’t get the usual “nonrelativistic string” as a limit. For example, the excitations – left-moving waves – are still propagating by the speed of light along the long relativistic string, even if they’re small. But they are moving by much smaller speeds, determined by the material constants, on the piano string.
I feel that the fluxtube is a relativistic string and this will give you a different velocity of the excitations along the long string, namely “c”, and your calculation using a nonrelativistic string just generates a wrong power law because the speed obtained as an intermediate result is much smaller than “c” which might be wrong. What do you think?
But maybe, you are assuming the speed of light for the waves along the fluxtube.
There was one more technical mistake I made in my previous thinking. If you have an open relativistic string stretched on distance R, between two branes, the mass goes like R times the tension, of course. But I was deluded in arguing what the excitations add. I thought they were adding sqrt(1/alpha’) to the mass.
That’s, of course, wrong. They are adding 1/alpha’ or so to the squared mass, which is something different because you don’t start at zero. The mass is changed into (in string units)
sqrt(R^2 + 1) = R + 1/2R + …
So the mass gap above the stretched string goes like the inverse separation (divided by alpha’). That might be what you get for your “nonrelativistic” string, too, unless you make it brutally nonrelativistic by making the ratio of tension and mass-per-unit-length brutally (parameterically) different from one.
Dear Lubos
Well, being completely accurate, QG tube is not quite the same thing as the flux line. There is QG plasma in the tube (of course, strongly coupled), so some kind of material constants should be also present in this case. For example, effective equation of state for the QG plasma may be lower than 1/3, so the speed of sound will be lower than c/\sqrt{3} (and c/\sqrt{3} is already a bit lower than c).
Could be very well as you explained, and I am missing something important. I have the following picture in mind. Quarks are infinitely heavy, their position is fixed (they are therefore made as non-relativistic as possible
). Then, what we have is just static interquark potential between them, V=\sigma L. Thinking in terms of this potential, I do not need to remember about the string exactly in the same way as I don’t need to think about photons propagating between heavy nuclei and electrons in atoms.
If I return the tension of the string you omitted back into the Eq., it seems that your mass gap does not depend on the tension (i.e., on alpha’). How can it be so? (suppose, I sent the tension to 0).
Anyway, I do feel too now, that the string should be always relativistic (apart from the fact that relativistic string is seen in N=\infty lattice QCD). I just would like to understand the subject better.
Cheers
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