Hopf semimetals are 4D gapless phases constructed via unstable homotopy maps from T^3 to S^2 that host nodal lines carrying Hopf flux along with unique gapless Fermi-arc, drumhead, Fermi-surface, and corner states.
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Direct observation of gapless moiré-Dirac quasiparticles forming topological nodal lines protected by non-symmorphic symmetry, with control via moiré periodicity.
The number of sheets and monodromy of the electron dispersion covering of the Brillouin zone in superstructured crystals are topological invariants, realized in three-sublattice helimagnet models as a spin-textured one-sheeted Fermi surface in a topological metal.
Two anomalous acoustic plasmons arise from the geometry of 2D type-II Dirac cones, beyond the conventional sqrt(q) form, with valley-dependent chirality and tunability by gap and substrate.
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Topological bands in metals
The number of sheets and monodromy of the electron dispersion covering of the Brillouin zone in superstructured crystals are topological invariants, realized in three-sublattice helimagnet models as a spin-textured one-sheeted Fermi surface in a topological metal.