  An analytical and numerical study of the one-dimensional double and
super-exchange model is presented. A phase separation between ferromagnetic and
anti-ferromagnetic phases occurs at low super-exchange interaction energy. When
the super-exchange interaction energy gets larger, the conduction electrons are
self-trapped within separate small magnetic polarons. These magnetic polarons
contain a single electron inside two or three sites depending on the conduction
electron density and form a Wigner crystallization. A new phase separation is
found between these small polarons and the anti-ferromagnetic phase. Our
results could explain the spin-glass-like behavior observed in the nickelate
one-dimensional compound Y(2-x)CaxBaNiO5.
