I was reading the review on Unconventional superconductivity by Mike Norman, towards the end (page 22) he comments two things about AdS/CMT:

  1. "In the condensed matter context in two dimensions, one typically flows from an $AdS_4$ geometry near the boundary to an $AdS_2$ times $R_2$ one near the black hole horizon. The net result is local quantum criticality, since the spatial $R_2$ part has essentially decoupled ($AdS_2$ being dual to $CFT_1$, a conformal field theory in time). In that sense, it is similar to the Kondo problem, which is local in space and critical in time."

  2. "Although this approach (AdS/CFT) is truly non-perturbative in nature, for most applications, the theory is essentially at the Ginzburg-Landau level. That is, one assumes a scalar field. Since it is a scalar, it should correspond to some charge 2e field, but since the theory does not explicitly invoke pairing, extra terms have to be added to the action to describe the coupling of fermions to the scalar field (i.e, to generate a Bogoliubov dispersion)."

I had hard time trying to understand the boldfaced statements, if someone can comment that'll be helpful. Thanks!

  • $\begingroup$ 1) It can be seen from the correlators for the operators in the dual field theory of AdS2 times R2, e.g. arxiv.org/abs/1105.4621 $\endgroup$ – user9510 Feb 23 '16 at 11:31
  • $\begingroup$ 2)The thermal phase transition in AdS/CFT is similar to GL theory, i.e. it is a mean field phase transition. However, AdS/CFT of course goes beyond GL theory, see e.g. arxiv.org/abs/1110.3814 $\endgroup$ – user9510 Feb 23 '16 at 11:33

I don't know much about the AdS/CFT stuff, but the statements about the Kondo effect and superconductivity are rather straightforward:

1) The classical setting for the Kondo effect is an impurity spin coupled to a Fermi sea, say e.g. a 1D fermion system, and the interaction between the spin and the fermions results in a nontrivial fixed-point for the boundary condition of fermions. So the effect is in a sense a boundary one, and one can study it by integrating out the bulk fermions. So it is extremely local in space (just one point), and critical in time (meaning that at the fixed point, if we look at spin-spin correlation function in time it falls off following a power law).

2). The Ginzburg-Landau level means that the superconductivity is directly described by a charge-$2e$ scalar field, the order parameter, ignoring the electronic origin. A truly microscopic theory, like the BCS theory, starts from electrons and phonons and derive the Ginzburg-Landau theory from there.

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    $\begingroup$ Thanks for the answer, particularly the Kondo one is helpful. However it's still not clear how can one draw analogy between these things and AdS/CFT. I'll wait for some AdS/CFT people to answer. $\endgroup$ – Jon Snow Dec 26 '15 at 0:20

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