Monday, February 15, 2010

"Holographic Superconductos in M-theory"

Last Monday we had Jerome Gauntlett telling us about exciting applications of the AdS/CFT correspondence to Condensed Matter systems.

Some quantum critical points (second order phase transitions) are at strong coupling, and hence intractable with standard CM technics, but the AdS/CFT comes to the rescue. This happens for example for "heavy fermions" (quasi-particles condensing to give Cooper pairs) or for high $T_c$ cuprates that have a superconducting phase. Holographic descriptions of such systems have been put forth by Gubser and by Hartnoll, Herzog, and Horowitz. The strongly coupled CFT in 3d has a holographic AdS dual, which has a $U(1)$ gauge symmetry that can be spontaneously broken by a charged field $\chi$. A finite temperature the description is in terms of a black hole in AdS, and it acquires charged hair at low temperature that breaks the $U(1)$.

Almost all the work on this topic has been taking a "bottom up" approach, but Jerome took the "top down" approach: constructing holographic superconductors from M-theory using consistent Kaluza-Klein truncation with Sasaki-Einstein spaces. Here consistent truncation means that you can keep only a finite set of the light KK fields (you cut the infinite KK towers) but then any solution of the low dimensional theory involving those remaining fields has to uplift to an exact solution of the original higher dimensional theory. So consider a stack of M2-branes at the tip of a Calabi-Yau cone on a Sasaki-Einstein seven-manifold for which the near-horizon limit is $AdS_4 \times SE_7$ with metric \[ d s^2 = \frac{1}{4} d s^2 (AdS_4) + d s^2 (SE_7) \] with \[ d s^2 (SE_7) = d s^2 (KE_6) + \eta \otimes \eta~,\] where $\eta$is the contact form. There is an associated Killing vector, which (although it does not always have to close) generates the required $U(1)$ symmetry. The four-form field strength is $G_4 = \frac{3}{8} \text{Vol}(AdS_4)$. This geometry is dual to $\mathcal{N} =3$ SCFTs in $d=3$ (or $\mathcal{N} = 8$ if $SE_7$ is replaced by a seven-sphere).
Now, had the membranes been replaced by anti-membranes, $G_4$ would have the opposite sign but more importantly the dual SCFTs would have $\mathcal{N}=0$. The corresponding geometry is called skew-whiffed $AdS_4 \times SE_7$. It is this geometry that can reproduce and generalize the HHH models. The consistent truncation found by Jerome has an extra neutral scalar field $h$ which corresponds to a deformation of the skew-whiffed CFT by an operator $\mathcal{O}_h$. At low temperature this system exhibits a superconducting phase.
Jerome concluded by suggesting that if superconductivity had not already been discovered, such a study would have pointed it out to us. M-theory might thus lead to a similarly radical discovery of qualitatively new phenomena...