  Employing a combination of first-principles calculations and low-energy
effective models, we present a comprehensive investigation on the electronic
structure of Pb$_{10}$(PO$_{4}$)$_{6}$O$_{4}$, which exhibits remarkable
quasi-one-dimensional flat-band around the Fermi level that contains novel
multi-dimensional fermions. These flat bands predominantly originate from
$p_x/p_y$ orbital of the oxygen molecules chain at $4e$ Wyckoff positions, and
thus can be well-captured by a four-band tight-binding model. Furthermore, the
abundant crystal symmetry inherent to Pb$_{10}$(PO$_{4}$)$_{6}$O$_{4}$ provides
an ideal platform for the emergence of various multi-dimensional fermions,
including a 0D four-fold degenerated Dirac fermion with quadratic dispersion, a
1D quadratic/linear nodal-line (QNL/LNL) fermion along symmetric $k$-paths, 1D
hourglass nodal-line (HNL) fermion linked to the Dirac fermion, and a 2D
symmetry-enforced nodal surface (NS) found on the $k_z$=$\pi$ plane. Moreover,
when considering the weak ferromagnetic order, Pb$_{10}$(PO$_{4}$)$_{6}$O$_{4}$
transforms into a rare semi-half-metal, which is characterized by the presence
of Dirac fermion and HNL fermion at the Fermi level for a single spin channel
exhibiting 100$\%$ spin polarization. Our findings reveal the coexistence of
flat bands, diverse topological semimetal states and ferromagnetism within in
Pb$_{10}$(PO$_{4}$)$_{6}$O$_{4}$, which may provide valuable insights for
further exploring intriguing interplay between superconductivity and exotic
electronic states.
