LAUSR

Recently, the notion of topological phases of matter has been extended to higher-order incarnations, supporting gapless modes on even lower dimensional boundaries, such as corners and hinges. We here identify a collection of cubic spin-$\frac{3}{2}$ fermions with biquadratic touching of Kramers degenerate valence and conduction bands as a platform to strain-engineer higher-order topological (HOT) phases: external uniaxial strain gives birth to a HOT Dirac semimetal or an insulator, depending on its sign, featuring topological hinge modes in the strain direction. The insulator in fact exhibits mixed topology, and in addition supports edge modes on orthogonal planes. These outcomes are germane when the external strain is applied along one of the ${C}_{4v}$ or coordinate axes, as well as ${C}_{3v}$ or body-diagonal, directions. Our findings place HgTe, gray-Sn, 227 pyrochlore iridates and half-Heusler compounds at the frontier of strain-engineered electronic HOT phases.