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AdS/CFT and condensed matter physics

Two months ago or so, we described how string theory is beginning to explain and take over portions of heavy ion physics:

The relevant description of many phenomena in heavy ion physics is governed by four-dimensional gauge theories that - as string theory teaches us - admit a dual, holographic description in terms of five-dimensional Anti de Sitter space (AdS5). The canonical theory studied in this context is SU(N), N=4 four-dimensional gauge theory (CFT4), the most popular field theory among a very large portion of theoretical physicists in 2007.

Several heavy ion physicists and their string-theoretical colleagues have realized that this simple gravitational theory in AdS5 describes many observations at RHIC and elsewhere in terms of black holes and other objects from string theory remarkably accurately and the underlying mathematics is unexpectedly simple and natural. There exist contexts where the maximally supersymmetric theory gives misleading results but the common assumption is that they can be described by a different, less symmetric background of string theory.

Many physicists - string theorists as well as heavy ion physicists - actually study some points in the "landscape" of solutions of string theory. Please, don't get irritated. When I say "landscape", it doesn't yet mean that I defend the anthropic principle or something of this kind. ;-)



Superfluids and insulators

Prof Subir Sachdev, a condensed matter physicist, had a very natural, simple, yet ingenious idea. If string theory is taking over heavy ion physics, shouldn't we expect that it will take over parts of condensed matter physics, too? As they figured out in a paper with Herzog, Kovtun, and Son, the answer is probably Yes. Subir just gave an exciting talk whose title coincided with the title of their new preprint:
One half of the audience were condensed matter physicists, the other half were string theorists.




Instead of the AdS5/CFT4 duality employed in heavy ion physics, they study a system that uses the AdS4/CFT3 duality. Just like string/M-theory contains many backgrounds whose geometry is a product of AdS5 and a compact space, it also predicts the existence of many backgrounds whose geometry is AdS4 times something.

The canonical and most symmetric example in this case is the N=8 superconformal field theory in 2+1 dimensions that is dual to the "AdS4 x S7" near-horizon geometry of M2-branes in 11-dimensional M-theory. The 2+1-dimensional theory can be constructed as the long-distance limit of the 2+1-dimensional maximally supersymmetric gauge theory describing D2-branes in type IIA string theory. The long-distance limit is equivalent to a strong coupling limit of the type IIA string theory - something that leads us to M-theory with a new, eleventh dimension and that transforms D2-branes to M2-branes. The R-symmetry is promoted from SO(7) to SO(8): the additional eighth adjoint scalar arises from the Hodge dual of the electromagnetic field. The enhancement of the R-symmetry follows from supersymmetry but it is much easier to understand it via the string/M-theoretical realization of the field theories.

This CFT with 16 supercharges and 16 additional superconformal generators plays the same important role for the stringy analysis of 2+1-dimensional systems such as those in condensed matter physics as the N=4 gauge theory in 3+1 dimensions played for situations such as heavy ion physics. It is the most symmetric, most easily calculable, and most well-established example of a theory in this universality class. Other theories or string theory backgrounds are less symmetric, less calculable, but they still share the same properties. At least qualitatively.

Time to get to real physics

So what are the physical systems from the real world that Subir et al. want to investigate? They want to look at models analogous to the boson Hubbard model in 2+1 dimensions, a canonical textbook material for condensed matter physicists. This system exhibits a very interesting phase transition between an insulator and a superfluid.

Take a thin foil of bismut whose thickness is X. Keep X fixed but send the temperature T to zero. What will happen with the conductivity? It turns out that for a vanishing temperature, the conductivity goes either to zero or to infinity. There must exist a critical thickness in between these two regimes where the conductivity remains finite. As you could expect, such phase transitions are controlled by a conformal field theory. The class of conformal field theories that you can obtain in these situations are called the Wilson-Fisher fixed points.

In the condensed matter approach, the 2+1-dimensional superconformal theory describing M2-branes is a fancy generalization of the Wilson-Fisher fixed point.

The relevant quantity that you want to measure or compute is a correlator of two conserved currents "J_mu^a(x,y,t)". Take the retarded correlator and find its Fourier transform. You will obtain a function of the frequency and the wavenumber that depends on four indices. The dependence on the Lorentz vector indices "mu,nu" splits the result into a transverse tensor and a longitudinal tensor. Schematically,
  • [ J J ] = KL(omega,k) LongiTensor + KT(omega,k) TransTensor
The LongiTensor and TransTensor are completely fixed ordinary tensors that depend on the "omega,k" vector in the standard way while the coefficients "KL,KT" could a priori be arbitrary functions of the frequency and the wavenumber. In 1+1 dimensions, there would be rather easy relations between all the analogous quantities, because of the holomorphic factorization of two-dimensional CFTs. However, in 2+1 dimensions, things could be harder and they are harder. However, Sachdev et al. show that
  • KL(omega,k) KT(omega,k) = N^3 / 72 pi^2
for the toy model of "N" M2-branes. The coefficients in the correlator - something encoding the conductivity - are inverse to each other. The non-trivial message here is that the product doesn't depend on "omega,k".

This self-duality has been viewed as a consequence of a particle-vortex duality in the past. However, Sachdev et al. derive it much more quantitatively and naturally from the AdS/CFT correspondence. The duality between the longitudinal modes and the transverse mode can be reduced to an S-duality in a limit of a dual gauge theory in the bulk.

Recall that the dual bulk description involves an SO(8) R-symmetry in the bulk arising from the isometry of the seven-sphere if we view it as a four-dimensional theory defined on AdS4. If the theory is weakly coupled, it is very close to a U(1)^{28} Abelian gauge theory - because 28 is the dimension of SO(8) - and one can apply four-dimensional S-duality on all these U(1) factors. This symmetry manifests itself as the "holographic self-duality", as they call it, or a particle-vortex duality in the three-dimensional boundary CFT description.

Although the actual two-spatial-dimensional materials they could construct are a bit different, so far there have been no really convenient and natural methods to calculate quantities in the hydrodynamical limit of these 2+1-dimensional theories. The dual gravitational description controlled by a large non-evaporating AdS4 Schwarzschild black hole seems to do the job very well: it kind of trivializes the situation.

Andy Strominger and myself were kind of surprised by Subir's statement that he could have obtained exactly Lorentz-invariant interacting theories in the infrared by adjusting one parameter only in a non-relativistic theory with particles and holes. Some of us have asked the question whether it is possible to create materials in the lab that would behave as a supersymmetric conformal theory in 2+1 dimensions. The answer was essentially No.

Nevertheless, it is becoming clear that the mathematical concepts and manipulations that were first discovered in the context of very high energy physics are recycled at many places of Nature, as Joe Polchinski puts it. It is a good opportunity for every physicist (and other scholar) to see whether
they can improve their understanding by applying a well-established method of string theory - such as the AdS/CFT correspondence - to their subject. The scientists who encounter conformal, scale-invariant theories are the first ones who should try to look what string theory can teach them.

And that's the memo.

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reader Mauricio said...

Lubos, I believe that the relation between heavy ion collisions and string theory is not totally clear, last article of Blaizot et al. show the differences of real calculations between AdS/CFT correspondence and other scenarios derived from thermal field theory. From the experimental point of view, there are no agreement between experiments realized in RHIC and string theory predictions, you can see for example many of the talks given in QM06. Moroever, there are many limitations when the people say that N=4 SYM looks like QCD at high temperatures, for example, you can see for example the problems concerning with this approach given by Liu in QM 06. I believe that possible maybe this problems can be solved in some future, but at the moment, it's a little exaggerated to claim that string theory is explaining portions of heavy ion physics


reader Lumo said...

Dear Mauricio, I suppose you mean hep-th/0611393.

I would expect a little bit more modesty about this paper with 1 citation currently.

Could you explain what's wrong with the standard papers about the subject such as hep-th/0605182 that have about 50 citations? It is these papers that are referred to by collaborations of hundreds of experimenters (for giving a correctly low value of D_{HQ}) among other things, see e.g.

nucl-ex/0611018.

Do you agree that I have thoroughly debunked your weird speculations? Let's time show what's right and what's not but your attempts to support your strange claims sociologically have completely failed.

Thanks for your understanding, Lubos


reader Mauricio said...

Dear Lubos:
McLerran explains in the last section of his last paper the main problem (hep-ph/0702004), that N=4 supersymmetric Yang-Mills is a quite different theory than QCD:

"Even in lowest order strong coupling computations it is very speculative to make relationships between this theory and QCD, because of the above. It is much more difficult to relate non-leading computations to QCD… The AdS/CFT correspondence is probably best thought of as a discovery tool with limited resolving power. An example is the eta/s computation. The discovery of the bound on eta/s could be argued to be verified by an independent argument, as a consequence of the deBroglie wavelength of particles becoming of the order of mean free paths. It is a theoretical discovery but its direct applicability to heavy ion collisions remains to be shown."

This is part of my doubts concerning with the application of AdS/CFT correspondence to heavy ion collisions, moroever, you say that the article of Liu have 50 citations meanwhile Blaizot's article just 1, more than 75% of the citations of Liu's article comes from people working in string theory, moroever this article is older than other. Furthermore, I am absolutely convinced that the community of string theory is nowadays bigger that heavy ion physics community, so for simple probability it will be easier if some article from string theory side has more citations than some article from heavy ions...Now, the experimental article are not giving any kind of agreement between string theory and the experiment, just comment that some parameter was calculated using AdS/CFT correspondence. Is this some agreement between experiment and string theory? I am not totally convinced about that, more research should be done with the aim to get better and more clear results. Lubos, I don't claim that string theory is bad, I just tell that the relation between heavy ions and string theory is not understood and there are many unclear hypothesis from string theory in this connections (see for example many observations given in Gubser's article hep-th/0611005)


reader Lumo said...

Dear Maurizio,

QCD is a different theory from N=4 Yang-Mills in general, but one of the major discoveries of the last few years in this field is that in the heavy ion processes, the gauge theories have a universal behavior and the difference is very small and N=4 theory simply *is* a good starting point to describe viscosity of QGP etc. whether someone likes it or not.

Dr. Melerran or what's his name has missed a "couple" of results which is why his comments are nothing else than confusion.

I personally don't assign too much weight to citations but your attempts to blow up 1 citation because of ideological reasons above 50 is just painful. What's exactly wrong with the people who study it being string theorists? Of course that the best conceptually important papers on heavy ion physics will probably be written by string theorists because string theorists got interested in this field and what we call "string theorists" is simply the same group that we could also call "the best theoretical physicists the world has".

It is completely irrelevant whether someone labels members of this group "string theorists" or "quantum gravity theorists" or "theoretical nuclear physicists" or "high-energy theorists" or anything else. It's the same people and the only thing that matters is the content of the papers.

Your attempts to create barriers between communities is shameful. There are no barriers in theoretical physics. A paper in theoretical physics is either correct or wrong, important or unimportant, or something in between, and judging papers according to some hypothetical membership of the authors in various groups is a disgusting attempt to politicize science.

Could you please read the papers about the subject regardless of your preconceptions about the races, genders, nationality, or other group affiliation of the authors? Thanks. Let me tell you that this is the ultimate minimum of moral standards you are required to respect if you want to post on this blog.

Best
Lubos


reader Mauricio said...

Dear Lubos,
I believe that this discussion takes another way different from which I had at the beginning.
I don't have any kind of ideological questions (races, religions whatever you want) and I didn't start any kind of barriers between different groups, but well, it was your interpretation. Criticism is important in science, and I'm not totally as a physicist in many statements from string theory side (many other people has made, for example, Glashow, Weinberg and others). I hope that future experiments can show which theory is right, nowadays, string theory doesn't have any experimental evidence as many other models.