  Inspired by recent experiments with cold atoms in optical lattices, we
consider a St\"uckelberg interferometer for a particle performing Bloch
oscillations in a tight-binding model on the honeycomb lattice. The
interferometer is made of two avoided crossings at the saddle points of the
band structure (i.e. at M points of the reciprocal space). This problem is
reminiscent of the double Dirac cone St\"uckelberg interferometer that was
recently studied in the continuum limit [Phys. Rev. Lett. 112, 155302 (2014)].
Although the two problems share similarities -- such as the appearance of a
geometric phase shift -- lattice effects, not captured by the continuum limit,
make them truly different. The particle dynamics in the presence of a force is
described by the Bloch Hamiltonian $H(\boldsymbol{k})$ defined from the
tight-binding Hamiltonian and the position operator. This leads to many
interesting effects for the lattice St\"uckelberg interferometer: a twisting of
the two Landau-Zener tunnelings, saturation of the inter-band transition
probability in the sudden (infinite force) limit and extended periodicity or
even non-periodicity beyond the first Brillouin zone. In particular,
St\"uckelberg interferometry gives access to the overlap matrix of
cell-periodic Bloch states thereby allowing to fully characterize the geometry
of Bloch states, as e.g. to obtain the quantum metric tensor.
