Hopf Semimetals: Protected Gapless Phases from Unstable Homotopies*

We construct and explore two-band topological semi-metals in different spatial dimensions that are protected by unstable homotopies. These semimetals that generically host nodal lines are dubbed "Hopf semimetals'', inspired by the example of such phases realized in four dimensions arising from maps from the three-torus T³ (Brillouin zone of a 3D crystal) to the two-sphere S² . In the four-dimensional example, a surface enclosing such a nodal line in Brillouin zone carries a Hopf flux. These 4D semimetals show a unique class of surface states: while some three-dimensional surfaces host gapless Fermi-arc states and drumhead states, other surfaces have gapless  Fermi surfaces. Gapless two-dimensional corner states are also present at the intersection of three-dimensional surfaces. We also demonstrate such semimetals realized in three dimensions in chiral class AIII, that arise from the unstable homotopies of maps from T² (Brillioun zone of a 2D crystal) to S¹. These 3D semimtals also hosts nodal lines, accompanied by a rich collection of surface states including drumhead type. This work provides a new framework to realize protected nodal line semimetals particularly in synthetic quantum systems such as cold atoms and photonic systems.