Dirac cones lie at the heart of many interests in studying two-dimensional crystals such as
graphene systems. Besides its unusual density-of-states, a Dirac cone possesses geometric
information hidden in the wavefunction, e.g., Berry's phase and mass. The latter determines
the type of topological states of matter. Thanks to recent progresses in engineering topological
band structure with cold atoms, the physical parameter regime in artificial crystals extends
beyond that of its solid-state counterpart. For example, the merging of Dirac cones, a
topological Lifshitz transition, was first experimentally realized in cold atoms and probed via
Landau-Zener transitions [1]. We provide a quantitative model to account fully for the
signatures observed [2]. When considering successive Landau-Zener transitions with two Dirac
cones, a Stückelberg interferometer is realized. We show that the interference pattern contains
an extra phase shift in the form of a gauge-invariant open-path Berry phase of the
wavefunction [3]. The latter is a Bloch wave analogue of the Pancharatnam's phase in optical
interference, in that it reveals band coupling effect of quantum crystals. The latter can serve as
a useful bulk probe for topological band structures.

References:
[1] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, Nature 483, 302 ( 2012).
[2] L.-K. Lim, J.-N. Fuchs, and G. Montambaux, Phys. Rev. Lett. 108, 175303 (2012).
[3] L.-K. Lim, J.-N. Fuchs, and G. Montambaux, Phys. Rev. Lett. 112, 155302 (2014).