Non-Abelian Euler topology in ultracold atoms

Topological insulators (TIs) are gapped quantum phases that have a topological nature by virtue of protecting symmetries. Following time reversal symmetry (TRS) protected TIs, past years have seen remarkable progress in characterizing topological materials taking into account crystal symmetries. These pursuits also revealed fragile invariants that can be trivialized by gap closings with trivial bands, rather than involving those having opposite topological charge. An archetypal novel invariant emerging from such studies, which goes beyond symmetry eigenvalue indicated phases and relates to refined partitioning schemes is Euler class. It acts as the fragile crystalline-protected analogue of Chern number in systems having C₂T [product of two-fold rotations and TRS] or PT [product of parity P and TRS] symmetry. Moreover, band nodes in Euler class models have charges that have non-Abelian braiding properties, enabling the study of non-Abelian physics essentially in a single particle setting and making their experimental realization highly desirable. On another front, ultracold atomic gases have proven versatile platforms for exploring topological phenomena with particular advances in periodic driving and artificial gauge fields, calling for expansion of these notions to out of equilibrium and bringing new classification schemes. In light of recent experiments calibrated to explore Euler topology, we here investigate the dynamics of Euler class in optical lattices. We study non-Abelian Euler properties in different out-of-equilibrium settings and identify unique observable signatures.