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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 stronglycorrelated materials, such as HgTe, $\alpha$-Sn, and iridate compounds.Previously, we have theoretically demonstrated that the Luttinger semimetalexhibits surface states. In the present work, we theoretically study theevolution of these surface states under symmetry-lowering perturbations:compressive strain and bulk-inversion asymmetry (BIA). This system is quitespecial in that each consecutive perturbation creates a new type of a semimetalphase, resulting in a sequence of four semimetal phases, where each successivephase 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. Wecalculate the surface states within the generalized Luttinger model for thesefour semimetal phases within a ``semi-analytical'' approach and fully analyzethe corresponding evolution of the surface states. Importantly, for thissequence of four semimetal phases, there is a corresponding hierarchy of thelow-energy models describing the vicinities of the nodes. We derive most ofthese models and demonstrate quantitative asymptotic agreement between thesurface-state spectra of some of them. This proves that the mechanismsresponsible for the surface states are fully contained in the low-energy modelswithin their validity ranges, once they are supplemented with proper boundaryconditions, and demonstrates that continuum models are perfectly applicable forstudying surface states.

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