Evolution of the surface states of the Luttinger semimetal under strain and inversion-symmetry breaking: Dirac, line-node, and Weyl semimetals

The Luttinger model of a quadratic-node semimetal for electrons with the $j=\frac32$ angular momentum under cubic symmetry is the parent, highest-symmetry low-energy model for a variety of topological and strongly correlated materials, such as HgTe, $\alpha$-Sn, and iridate compounds. Previously, we have theoretically demonstrated that the Luttinger semimetal exhibits surface states. In the present work, we theoretically study the evolution of these surface states under symmetry-lowering perturbations: compressive strain and bulk-inversion asymmetry (BIA). This system is quite special in that each consecutive perturbation creates a new type of a semimetal phase, resulting in a sequence of four semimetal phases, where each successive phase arises by modification of the nodal structure of the previous phase: under compressive strain, the Luttinger semimetal turns into a Dirac semimetal, which under the linear-in-momentum BIA term turns into a line-node semimetal, which under the cubic-in-momentum BIA terms turns into a Weyl semimetal. We calculate the surface states within the generalized Luttinger model for these four semimetal phases within a ``semi-analytical'' approach and fully analyze the corresponding evolution of the surface states. Importantly, for this sequence of four semimetal phases, there is a corresponding hierarchy of the low-energy models describing the vicinities of the nodes. We derive most of these models and demonstrate quantitative asymptotic agreement between the surface-state spectra of some of them. This proves that the mechanisms responsible for the surface states are fully contained in the low-energy models within their validity ranges, once they are supplemented with proper boundary conditions, and demonstrates that continuum models are perfectly applicable for studying surface states.