  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.
