Flat-band and multi-dimensional fermions in Pb10(PO4)6O4

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.