  Topological insulators (TIs) and Weyl semimetals (WSMs) are two realizations
of topological matter usually appearing separately in nature. However, they are
directly related to each other via a topological phase transition. In this
paper, we investigate the question whether these two topological phases can
exist together at the same time, with a combined, hybrid surface state at the
joint boundaries. We analyze effective models of a 3D TI and an inversion
symmetric WSM and couple them in a way that certain symmetries, like inversion,
are preserved. A tunnel coupling approach enables us to obtain the hybrid
surface state Hamiltonian analytically. This offers the possibility of a
detailed study of its dispersion relation depending on the investigated
couplings. For spin-symmetric coupling, we find that two Dirac nodes can emerge
out of the combination of a single Dirac node and a Fermi arc. For
spin-asymmetric coupling, the dispersion relation is gapped and the former
Dirac node gets spin-polarized. We propose different experimental realization
of the hybrid system, including compressively strained HgTe as well as
heterostructures of TI and WSM materials.
