EuB$_6$ and the Low-Density Double Exchange Model

The low electronic density regime of the double exchange model is explored in the presence of electron-electron interactions. The single particle problem and its extension to low densities, when a Wigner crystal of magnetic polarons is generated due to unscreened Coulomb interactions, is studied. It is argued that the Wigner crystal is the natural alternative to phase separation when the Coulomb interaction is taken into account. We address the thermal and quantum stability of the crystalline phase towards a polaronic Fermi liquid and a homogeneous, metallic, ferromagnetic phase. These results are relevant in the context of the polaronic physics recentely observed in EuB$_6$, and provide an important consistency-check for the double-exchange description of these magnetic hexaborides. \newline {\bf References:} \newline V.~M.~Pereira \emph{et~al.}, \texttt{ cond-mat/0505741}; \newline V.~M. Pereira \emph{et~al.}, Phys. Rev. Lett. \textbf{93}, 147202 (2004).