Classification And Signatures of Multi-gap Band Topologies Far From Equilibrium

While the landscape of free-fermion phases has drastically been expanded in the last decades, recently novel multi-gap topological phases were proposed where groups of bands can acquire new invariants such as Euler class. As in conventional single-gap topologies, obstruction and dynamics play an inherent role and enrich the multi-gap topology which so far have been only incidentally addressed. We present a comprehensive theoretical classification to capture the obstruction effects on the multi-gap topology of Bloch systems and illuminate the out-of-equilibrium-signatures that arise as a consequence, which we also verified numerically. Our work presents a general framework to study the role of lattice configurations and obstructions on multi-gap topology that can be employed for arbitrary number bands or in higher dimensions and in out-of-equilibrium settings.