  We explore the topological properties of double-Weyl semimetals with cold
atoms in optical lattices. We first propose to realize a tight-binding model of
simulating the double-Weyl semimetal with a pair of double-Weyl points by
engineering the atomic hopping in a three-dimensional optical lattice. We show
that the double-Weyl points with topological charges of \$pm2$ behave as sink
and source of Berry flux in momentum space connecting by two Fermi arcs and
they are stabilized by the \$C_{4h}$ point-group symmetry. By applying a
realizable \$C_4$ breaking term, we find that each double-Weyl point splits
into two single-Weyl points and obtain rich phase diagrams in the parameter
space spanned by the strengths of an effective Zeeman term and the \$C_4$
breaking term, which contains a topological and a normal insulating phases and
two topological Weyl semimetal phases with eight and four single-Weyl points,
apart from the double-Weyl semimetal phase. Furthermore, we demonstrate with
numerical simulations that (i) the mimicked double- and single-Weyl points can
be detected by measuring the atomic transfer fractions after a Bloch
oscillation; (ii) the Chern number of different quantum phases in the phase
diagram can be extracted from the center shift of the hybrid Wannier functions,
which can be directly measured with the time-of-flight imaging; (iii) the band
topology of the \$C_4$ symmetric Bloch Hamiltonian can be detected simply from
measuring the spin polarization at the high symmetry momentum points with a
condensate in the optical lattice. The proposed system would provide a
promising platform for elaborating the intrinsic exotic physics of double-Weyl
semimetals and the related topological phase transitions.
