I'm going to be teaching a course on gauge/gravity duality (aka AdS/CFT) in the winter. The focus will be on applications in particle theory including $N=4$ SYM, the viscosity/entropy bound, and aspects of large $N$ QCD. I would like to include at least one application to condensed matter physics. There has been a lot of work on this over the last couple of years but I haven't followed it very closely. Can anybody suggest a nice application that is simple enough to work through in a few lectures for students who know QFT and GR and will have had some exposure to AdS/CFT?
I would suggest looking at Sachdev's "Condensed Matter and AdS/CFT" and McGreevy's "Holographic duality with a view toward many-body physics" for an overall perspective.
When it comes to condensed matter, AdS/CFT is generally used to calculate transport coefficients for systems near their critical point such as in the antiferromagnetic (Neel state) - dimer gas crossover or for high-Tc superconductivity to understand the transition between psuedogap and superconducting phases. Again for a more precise statement of the various cases I would look at these and other references.
Their is also a nice post on Lubos Motl's blog regarding Sachdev's work.
Jeff, holographic superconductivity/superfluidity is fairly simple conceptually and mechanically. The basic physics is breaking spontaneously an Abelian global symmetry. After you translate that to AdS language, this has to do with the existence of solutions to certain differential equations, with certain boundary conditions. The solutions can be constructed numerically (or in some cases analytically), but even without an explicit solution there are some arguments to demonstrate that such solutions exist. The first paper in the series by Horowitz, Hartnoll and Herzog may be a good one to follow.
It's simple to compute optical conductivity in 1/2 a lecture. It's not that exciting, but it is nice to see how you can use high powered techniques to derive something as simple as Ohm's law.
If you want to impress condensed matter types, computing the fermion spectral function and seeing marginal Fermi liquid behaviour is probably the best way to go.