  We propose a model of a strongly-interacting two-impurity Kondo system based
on the AdS/CFT correspondence, also known as holography. In a Landau Fermi
Liquid, the single-impurity Kondo effect is the screening of an impurity spin
at low temperature $T$. The two-impurity Kondo model then describes the
competition between the Kondo interaction and the Heisenberg interaction
between two impurity spins, also called the RKKY interaction. For spin-$1/2$
impurities, that competition leads to a quantum phase transition from a
Kondo-screened phase to a phase in which the two impurity spins screen one
another. Our holographic model is based on a $(1+1)$-dimensional CFT
description of the two-impurity Kondo model, reliable for two impurities with
negligible separation in space. We consider only impurity spins in a totally
anti-symmetric representation of an $SU(N)$ spin symmetry. We employ a
large-$N$ limit, in which both Kondo and RKKY couplings are double-trace, and
both Kondo and inter-impurity screening appear as condensation of single-trace
operators at the impurities' location. We perform the holographic
renormalization of our model, which allows us to identify the Kondo and RKKY
couplings as boundary conditions on fields in AdS. We numerically compute the
phase diagram of our model in the plane of RKKY coupling versus $T$, finding
evidence for a quantum phase transition from a trivial phase, with neither
Kondo nor inter-impurity screening, to a non-trivial phase, with both Kondo and
anti-ferromagnetic inter-impurity screening. More generally we show, just using
$SU(N)$ representation theory, that ferromagnetic correlations must be absent
at leading order in the large-$N$ limit. Our holographic model may be useful
for studying many open problems involving strongly-interacting quantum
impurities, including for example the Kondo lattice, relevant for describing
the heavy fermion compounds.
