  The discoveries of Dirac and Weyl semimetal states in spin-orbit compounds
led to the realizations of elementary particle analogs in table-top
experiments. In this paper, we propose the concept of a three-dimensional
type-II Dirac fermion and identify a new topological semimetal state in the
large family of transition-metal icosagenides, MA3 (M=V, Nb, Ta; A=Al, Ga, In).
We show that the VAl3 family features a pair of strongly Lorentz-violating
type-II Dirac nodes and that each Dirac node consists of four type-II Weyl
nodes with chiral charge +/-1 via symmetry breaking. Furthermore, we predict
the Landau level spectrum arising from the type-II Dirac fermions in VAl3 that
is distinct from that of known Dirac semimetals. We also show a topological
phase transition from a type-II Dirac semimetal to a quadratic Weyl semimetal
or a topological crystalline insulator via crystalline distortions. The new
type-II Dirac fermions, their novel magneto-transport response, the topological
tunability and the large number of compounds make VAl3 an exciting platform to
explore the wide-ranging topological phenomena associated with
Lorentz-violating Dirac fermions in electrical and optical transport,
spectroscopic and device-based experiments.
