Higher-order double-Weyl semimetal

Higher-order Weyl semimetal (HOWSM) is a fascinating topological phase that connects the nontrivial higher-order topology and Weyl semimetal. In this paper, we introduce a novel phase termed higher-order double-Weyl semimetals (HODWSMs) through the application of symmetry analysis and a minimal tight-binding model. This phase is notable for its quadratic double-Weyl nodes located in the $k_z=0$ plane and nontrivial higher-order topology in other planes. Specifically, within this phase, there is a sole pair of Weyl nodes, distinguished by chiral charges of $+2$ and $-2$, respectively. The emergence of these nodes can be attributed to the combined effect of $C_4$ rotation symmetry and time-reversal symmetry, whereas the higher-order topology stems from the filling anomaly. Through first-principles calculations, we propose that ${\mathrm{LaRhC}}_2$, an experimentally synthesized material, serves as a prototypical crystalline candidate for observing this unique HODWSM phase in its optic phonons. Additionally, further calculations demonstrate that the HODWSM phase exhibits direction-dependent surface and hinge states, providing distinctive features for future experimental detection of this intriguing and nontrivial topological phase.