AdS/CFT goes cold

Last week Dam Son gave two nice talks about phenomenological applications of AdS/CFT:
one about heavy ions, and the other about non-relativistic conformal field theories (CFTs). While the former application is widely discussed in pubs and blogs, the latter is a relatively new development. It seems that, after having entrenched in the heavy ion territory, particle theory has launched another offensive on the unsuspecting condensed matter folk. Not later than yesterday I saw two new papers on the subject posted on ArXiv.

AdS/CFT as we know it relates strongly coupled gauge theories to gravity theories in one more dimension. In the original tables received at Mount Sinai by Maldacena it speaks about highly symmetric and all-but-realistic theories: N = 4 super Yang-Mills on the gauge theory side and 10D type IIB supergravity in $AdS_5\times S_5$ background on the gravity side. Later, the correspondence was vulgarized to allow for phenomenological applications. In particular, some success was reported in postdicting meson spectra of low-energy QCD and explaining large viscosity of the quark-gluon plasma. Heavy ion collisions are total mess, however, and one would welcome an application in the field where the experimental conditions can be carefully tuned. Condensed matter physics enjoys that privilege and, moreover, laboratory systems near a critical point are often described by CFT. The point is that in most of the cases these are non-relativistic CFTs.

A commonly discussed example of a condensed matter system is the so-called fermions at unitarity (what's in the name?). This system can be experimentally realized as trapped cold atoms at the Feshbach resonance. Theoretically, it is described using a fermion field with the non-relativistic free lagrangian $\psi^\dagger \pa_t \psi - |\pa_x\psi|^2/2m$ and short range interactions provided by the four-fermion term $c_0 (\psi^\dagger \psi)^2$. The experimental conditions can be tuned such that $c_0$ is effectively infinite. In this limit the system has the same symmetry as the free theory and, in particular, it has scale invariance acting as $x \to \lambda x$, $t \to \lambda^2 x$. The full symmetry group includes also the non-relativistic Galilean transformations and special conformal transformations, and it is called the Schrodinger group (because it is the symmetry group of the Schrodinger equation). Most of the intuition from relativistic CFT (scaling dimensions, primary operators) carries over to the non-relativistic case.

The most important piece of evidence for the AdS/CFT correspondence is matching of the symmetries on both sides of the duality. For example, the relativistic conformal symmetry SO(2,4) of the SYM gauge theory in 4D is the same as the symmetry group of the AdS spacetime. In the case at hand we have a different symmetry group so we need a different geometric background on the gravity side. The Schrodinger group Sch(d) in d spatial dimensions can be embedded in the conformal group SO(d+2,2). For the interesting case d = 3 this shows that one should look for a deformation of the AdS background in six space-time dimensions, one more than in the relativistic case. In the paper from April this year, Dam Son identified the background with the desired symmetry properties. It goes like this
$ds^2 = \frac{-2 dx^+ dx^- + dx^i dx^i + dz^2}{z^2} - \frac{(dx^+)^2}{z^4}$.
The first term is the usual AdS metric, the last term is a deformation that reduces the symmetry down to the Schrodinger group Sch(d). The light-cone coordinate $x^-$ is compactified, and the quantized momentum along that coordinate is identified with the mass operator in the Schrodinger algebra.

So, the hypothesis is that fermions at unitarity have a dual description in terms of a gravity theory on that funny background. Many details of the correspondence are still unclear. One obstacle seems to be that fermions at unitarity do not have an expansion parameter analogous to the number of colors of relativistic gauge theories. A more precise formulation of the duality is clearly needed.