  The recent detailed study of quasi-one-dimensional iron-based ladders, with
the $3d$ iron electronic density $n = 6$, has unveiled surprises, such as
orbital-selective phases. However, similar studies for $n=6$ iron chains are
still rare. Here, a three-orbital electronic Hubbard model was constructed to
study the magnetic and electronic properties of the quasi-one-dimensional $n=6$
iron chain Ce$_2$O$_2$FeSe$_2$, with focus on the effect of doping.
Specifically, introducing the Hubbard $U$ and Hund $J_{H}$ couplings and
studying the model via the density matrix renormalization group, we report the
ground-state phase diagram varying the electronic density away from $n=6$. For
the realistic Hund coupling $J_{H}/U = 1/4$, several electronic phases were
obtained, including a metal, orbital-selective Mott, and Mott insulating
phases. Doping away from the parent phase, the competition of many tendencies
leads to a variety of magnetic states, such as ferromagnetism, as well as
several antiferromagnetic and magnetic "block" phases. In the hole-doping
region, two different interesting orbital-selective Mott phases were found:
OSMP1 (with one localized orbital and two itinerant orbitals) and OSMP2 (with
two localized orbitals and one itinerant orbital). Moreover, charge
disproportionation phenomena were found in special doping regions. We argue
that our predictions can be tested by simple modifications in the original
chemical formula of Ce$_2$O$_2$FeSe$_2$.
