  The physics of massless relativistic quantum particles has recently arisen in
the electronic properties of solids following the discovery of graphene. Around
the accidental crossing of two energy bands, the electronic excitations are
described by a Weyl equation initially derived for ultra-relativistic
particles. Similar three and four band semimetals have recently been discovered
in two and three dimensions. Among the remarkable features of graphene are the
characterization of the band crossings by a topological Berry winding, leading
to an anomalous quantum Hall effect, and a finite minimal conductivity at the
band crossing while the electronic density vanishes. Here we show that these
two properties are intimately related: this result paves the way to a direct
measure of the topological nature of a semi-metal. By considering three band
semimetals with a flat band in two dimensions, we find that only few of them
support a topological Berry phase. The same semimetals are the only ones
displaying a non vanishing minimal conductivity at the band crossing. The
existence of both a minimal conductivity and a topological robustness
originates from properties of the underlying lattice, which are encoded not by
a symmetry of their Bloch Hamiltonian, but by a duality.
