Delicate semimetals: Protected gapless phases from unstable homotopies

We construct and explore two-band topological semimetals in different spatial dimensions that are protected by unstable homotopies. Dubbed ``delicate semimetals,'' they generically host nodal lines and are inspired by the example of such phases realized in four dimensions arising from maps from the three-torus ${T}^{3}$ [Brillouin zone of a three-dimensional (3D) crystal] to the two-sphere ${S}^{2}$ related to the Hopf map. In the four-dimensional example, a surface enclosing such a nodal line in the Brillouin zone carries a Hopf flux. These four-dimensional semimetals show a unique class of surface states: while some 3D 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, which arise from the unstable homotopies of maps from ${T}^{2}$ (Brillouin zone of a two-dimensional crystal) to ${S}^{1}$. These 3D semimetals also host 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, photonic, and topoelectric systems.