  In this paper, a Bose-Hubbard extension of a Weyl semimetal is proposed that
can be realized for ultracold atoms using laser assisted tunneling and Feshbach
resonance technique in three dimensional optical lattices. The global phase
diagram is obtained consisting of a superfluid phase and various Mott insulator
phases by using Landau theory. The Bogoliubov excitation modes for the weakly
interacting case have nontrivial properties (Weyl nodes, bosonic surface arc,
etc.) analogs of those in Weyl semimetals of electronic systems, which are
smoothly carried over to that of Bloch bands for the noninteracting case. The
properties of the insulating phases for the strongly interacting case are
explored by calculating both the quasiparticle and quasihole dispersion
relation, which shows two quasiparticle spectra touch at Weyl nodes.
