  We consider ultracold atoms in a two-dimensional optical lattice of the dice
geometry in a tight-binding regime. The atoms experience a laser-assisted
tunneling between the nearest neighbour sites of the dice lattice accompanied
by the momentum recoil. This allows one to engineer staggered synthetic
magnetic fluxes over plaquettes, and thus pave a way towards a realization of
topologically nontrivial band structures. In such a lattice the real-valued
next-neighbour transitions are not needed to reach a topological regime. Yet,
such transitions can increase a variety of the obtained topological phases. The
dice lattice represents a triangular Bravais lattice with a three-site basis
consisting of a hub site connected to two rim sites. As a consequence, the dice
lattice supports three dispersion bands. From this point of view, our model can
be interpreted as a generalization of the paradigmatic Haldane model which is
reproduced if one of the two rim sub-lattices is eliminated. We demonstrate
that the proposed upgrade of the Haldane model creates a significant added
value, including an easy access to topological semimetal phases relying only on
the nearest neighbour coupling, as well as enhanced topological band structures
featuring Chern numbers higher than one. The numerical investigation is
supported and complemented by an analytical scheme based on the study of
singularities in the Berry connection.
