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The relatively unexplored family of pnictide-based antiperovskites has been shown to harbor prospects for topological phases. Using the example of ${\mathrm{Ca}}_{3}\mathrm{BiP}$, we demonstrate a cascade of phases as the initial cubic symmetry is broken successively. Initially a small-gap ${Z}_{2}$ topological insulator, spin-orbit coupling leads to band re-ordering resulting in a topological semimetal phase. Compressive uniaxial (001) strain leads back to a small-gap ${Z}_{2}$ topological insulator, with the expected gapless boundary modes. Tensile (001) strain leaves the system with a pair of Dirac points along $(0,0,\ifmmode\pm\else\textpm\fi{}{k}_{\ensuremath{\circ}})$ pinning the Fermi level, producing an unusual double meniscus of connected Fermi arcs on the (100) and (010) surfaces. Finally, breaking time-reversal symmetry by an applied Zeeman field produces a new phase with a pair of multi-Weyl nodes (massive or massless depending on direction, conventionally called semi-Dirac in two-dimensional systems) simultaneous with a sister pair of Dirac modes along each $\ifmmode\pm\else\textpm\fi{}{k}_{z}$ axis, combining to pin the Fermi level in the close vicinity of these varied single-particle excitations.

Coemergence of Dirac and multi-Weyl topological excitations in pnictide antiperovskites