![]() ![]() Here different colors denote different (co-)irreps of the little group of k c. The hourglass BC is denoted by a green dot which is at the crossing point of two neck bands in red and blue. (a) Left panel: the hourglass band structure along k c connecting k 1 and k 2. The hourglass band structures of types A–E are demonstrated in panels (a)–(e), respectively, where the band structures are depicted along k c connecting k 1 and k 2. ![]() The symmetry-guaranteed existence of hourglass fermion in the predicted magnetic materials is expected to be applied in manipulating band topology by applying appropriate external fields moving the hourglass BC. We take CsMn 2 F 6, synthesized recently with a distorted pyrochlore structure to illustrate the hourglass band structure in detail which is very clear around the Fermi level and the topologically protected surface drumhead states of (100) surface are found to spread over more than one half surface Brillouin zone and only appear in a narrow energy window ( ∼ 30 meV), which could induce intriguing stability by prominent electronic correlation. Among these results, the essential hourglass BCs are highlighted, whose MSGs are then applied to predict hundreds of magnetic materials from the MAGNDATA magnetic materials database and first-principles calculations in the frame of LDA + SOC + U verify the hourglass BCs for different values of U. Here we first list all symmetry conditions that allow hourglass BCs in the 1651 MSGs and 528 magnetic layer groups (MLGs) with spin-orbital coupling (SOC): Only 331 MSGs and 53 MLGs can host hourglass BCs. To date, there have been no magnetic material verified with hourglass fermions. Such novel property renders the theoretical prediction on magnetic topological metals with hourglass BC (which can be Weyl point, Dirac point, lying in nodal loop, and so on) independent on the calculation methods and only determined by the symmetry of crystal and magnetic structure, namely, the magnetic space group (MSG). Among various BCs, the so-called hourglass BCs (with the low-energy excitations dubbed as hourglass fermions) are fascinating since they can be guaranteed to exist under specific symmetry conditions even without realistic calculations. Many topological band crossings (BCs) have been predicted efficiently utilizing the symmetry properties of wave-functions at high-symmetry points. ![]()
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