Flat-band and diverse quasi-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 Pb₁₀(PO₄)₆O₄, which exhibits remarkable quasi-one-dimensional topological flat-band around the Fermi level. These flat bands predominantly originate from the px/py orbitals of the oxygen molecules chain at the fully-occupied 4e Wyckoff positions and thus can be well-captured by a minimal four-band tight-binding model. Furthermore, the abundant crystal symmetry inherent in Pb₁₀(PO₄)₆O₄ provides an ideal platform for the emergence of various quasi-fermions characterized by different dispersion, degeneracy, and dimensionality. These include a 0D four-fold degenerate Dirac fermion exhibiting quadratic dispersion, a 1D quadratic/linear nodal-line fermion along symmetric k-paths, a 1D hourglass nodal-line (HNL) fermion associated with the Dirac fermion, and a 2D symmetry-enforced nodal surface located on the k_z = π plane. Moreover, when considering the weak ferromagnetic order, Pb₁₀(PO₄)₆O₄ 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 rarely coexistence of flat bands, diverse topological semimetal states and ferromagnetism in Pb₁₀(PO₄)₆O₄, which may provide valuable insights for further exploring the intriguing interplay between superconductivity and exotic electronic states.