Strain-engineered higher-order topological phases for spin-$\frac{3}{2}$ Luttinger fermions

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.

Figure 1

Schematic phase diagram of a LSM (red dot), in the presence of an external strain of magnitude $\left|\mathrm{\Delta }\right|$. Here, $\mathrm{sgn}\left(\mathrm{\Delta }\right)=+(-)$ corresponds to tensile (compressive) strain, which gives rise to a second-order Dirac semimetal (topological insulator). The momentum in the direction of the external strain is $k_{\mathrm{str}}$. In the semimetal phase two Dirac points, located at $\pm k_{\mathrm{str}}^{*}$, are separated along this axis, where $k_{\mathrm{str}}^{*}\sim \sqrt{\left|\mathrm{\Delta }\right|}$. Both phases produce four hinge states of codimension $d_c=2$, along $k_{\mathrm{str}}$ [see blue lines in the cubes (inset)]. In the insulating (Dirac semimetal) phase the hinge modes exist for any $k_{\mathrm{str}}$ ($|k_{\mathrm{str}}|>k_{\mathrm{str}}^{*}$). The topological insulator also supports edge modes (green lines) with $d_c=1$ on the $k_{\mathrm{str}}=0$ plane (green dots), occupying two planes orthogonal to $k_{\mathrm{str}}$; thus accommodating mixed topology. The white region ($|k_{\mathrm{str}}|<k_{\mathrm{str}}^{*}$) is devoid of any boundary mode. In practice, $k_{\mathrm{str}}$ is a high-symmetry ($\langle 100\rangle$ or $\langle 111\rangle$, for example) direction.