Flat-band and diverse quasi-fermions in Pb₁₀(PO₄)₆O₄

Abstract 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 topological flat-band around the Fermi level. These flat bands predominantly originate from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>y</mml:mi> </mml:msub> </mml:mrow> </mml:math> orbitals of the oxygen molecules chain at the fully-occupied 4 e Wyckoff positions and thus can be well-captured by a minimal four-band tight-binding model. Furthermore, the abundant crystal symmetry inherent in Pb 10 (PO 4 ) 6 O 4 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, 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">%</mml:mi> </mml:mrow> </mml:math> spin polarization. Our findings reveal the rarely coexistence of flat bands, diverse topological semimetal states and ferromagnetism in Pb 10 (PO 4 ) 6 O 4 , which may provide valuable insights for further exploring the intriguing interplay between superconductivity and exotic electronic states.