  We investigate the magnetic behavior of nuclear spins embedded in a 2D
interacting electron gas using a Kondo lattice model description. We derive an
effective magnetic Hamiltonian for the nuclear spins which is of the RKKY type
and where the interactions between the nuclear spins are strongly modified by
the electron-electron interactions. We show that the nuclear magnetic ordering
at finite temperature relies on the (anomalous) behavior of the 2D static
electron spin susceptibility, and thus provides a connection between
low-dimensional magnetism and non-analyticities in interacting 2D electron
systems. Using various perturbative and non-perturbative approximation schemes
in order to establish the general shape of the electron spin susceptibility as
function of its wave vector, we show that the nuclear spins locally order
ferromagnetically, and that this ordering can become global in certain regimes
of interest. We demonstrate that the associated Curie temperature for the
nuclear system increases with the electron-electron interactions up to the
millikelvin range.
