  To analyze the physical properties arising from indirect magnetic exchange
between several magnetic adatoms and between complex magnetic nanostructures on
metallic surfaces, the real-space extension of dynamical mean-field theory
(R-DMFT) appears attractive as it can be applied to systems of almost arbitrary
geometry and complexity. While R-DMFT describes the Kondo effect of a single
adatom exactly, indirect magnetic (RKKY) exchange is taken into account on an
approximate level only. Here, we consider a simplified model system consisting
of two magnetic Hubbard sites ("adatoms") hybridizing with a non-interacting
tight-binding chain ("substrate surface"). This two-impurity Anderson model
incorporates the competition between the Kondo effect and indirect exchange but
is amenable to an exact numerical solution via the density-matrix
renormalization group (DMRG). The particle-hole symmetric model at half-filling
and zero temperature is used to benchmark R-DMFT results for the magnetic
coupling between the two adatoms and for the magnetic properties induced in the
substrate. In particular, the dependence of the local adatom and the nonlocal
adatom-adatom static susceptibilities as well as the magnetic response of the
substrate on the distance between the adatoms and on the strength of their
coupling with the substrate is studied. We find both, excellent agreement with
the DMRG data even on subtle details of the competition between RKKY exchange
and the Kondo effect but also complete failure of the R-DMFT, depending on the
parameter regime considered. R-DMFT calculations are performed using the
Lanczos method as impurity solver. With the real-space extension of the
two-site DMFT, we also benchmark a simplified R-DMFT variant.
