$n$-Hourglass Weyl fermions in nonsymmorphic materials

Hourglass-like band structures protected by nonsymmorphic space group symmetries can appear along high-symmetry lines or in high-symmetry surfaces in the Brillouin zone. In this work, from symmetry analysis, we demonstrate that $n$-hourglass-like band structures, a generalization of hourglass-like band structures, which host a number of Weyl points, are enforced along screw-invariant lines in non-magnetic materials with a single $N$-fold screw axis when spin-orbit coupling is finite, where $n$, a non-unity factor of $N$, denotes the degree of the screw-invariant line. The standard $n$-hourglass has $n-1$ crossings, which is minimal, and its variants can have more crossings. Purely by symmetry considerations, we find there are minimally two particle and two hole Fermi pockets enclosing Weyl points with opposite monopole charges at proper fillings, which can result in distinct physical effects including the possible formation of topological density waves and the quantum nonlinear Hall effect. We construct a minimal model which respects all the symmetries, and from which we see how the $n$-hourglasses appear when spin-orbit coupling is turned on. The same results are derived from compatibility relations. As exemplary, BiPd, ZnTe under high pressure, and the high-temperature phase of Tl$_3$PbBr$_5$, are shown from first-principles calculations to exhibit $n$-hourglass-like band structures, with $N=2,3,4$, respectively, which confirms our symmetry analysis and minimal model.