Observation of Topological Bloch-State Defects and Their Merging Transition

Abstract

Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudo-spin texture. The topological defects appear as vortices in the azimuthal phase of this pseudo-spin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.

Measuring topology from dynamics by obtaining the Chern number from a linking number

“…This robustness of topological defects is a general feature and was also used in the related work of ref. 9 . While the effect of dispersive bands of the initial Hamiltonian was discussed in refs.…”

“…Ultracold quantum gases are a promising experimental platform to explore these effects. On the one hand, they allow for the realization of topologically nontrivial band structures and artificial gauge fields 2–9 and on the other hand typical time scales for dynamical studies are experimentally accessible. Example, paradigmatic topological band models have been realized: the Hofstadter model describing a lattice with a net magnetic flux and the Haldane model on the honeycomb lattice, which contains topologically non-trivial bands even in the absence of a net magnetic flux.…”

Observation of sub-wavelength phase structure of matter wave with two-dimensional optical lattice by Kapitza-Dirac diffraction

“…Our scheme can be used to detect more delicate properties of the optical lattice, such as the fine structure within a unit cell and its associated flux [35][36][37] . Finally, the www.nature.com/scientificreports www.nature.com/scientificreports/ sub-wavelength phase structure in our work can also be connected to the topological defects in matter wave [38][39][40] , such as phase steps or vortices, which could be equally detected in the momentum space.…”

Criticality of Dirac fermions in the presence of emergent gauge fields

“…Lifshitz transitions of merging Dirac cones are observed when tuning through the broken-symmetry states close to the multicritical point. These insights could be relevant for a range of systems, including black phosphorus [80,81], optical honeycomb lattices [46,82,83], artificial graphene [84], TiO 2 /VO 2 interfaces [31,85], and α(BEDT-TTF) 2 I 3 [86].…”