Adrien Bouhon: Non-Abelian nodal topology and Euler topological insulators: from the homotopy groups of real Grassmannians and real flag manifolds to the topology and quantum geometry of crystalline many-band systems

par Adrien Bouhon

vendredi 27 mars 2026, 09:30 → 12:00 Europe/Paris

amphi C (ENS Lyon allée d'Italie)

I first review the homotopy classification of band structures for crystalline systems described by real Bloch Hamiltonians. These systems abound in nature as they merely require the combination of e.g. time reversal and a Pi-rotation symmetry axis. I introduce the notion of classifying space for real Bloch Hamiltonians, that are real Grassmannians and real flag manifolds. I then derive the systematic homotopy classification of topological phases in multi-band crystalline systems, showing that the rich topological structure of the classifying spaces leads to a great variety of topological phases in all dimensions. For this, I also introduce the topological invariants associated to given dimensionalities, e.g. non-Abelian homotopy charges of nodal points, the Euler class (of real vector bundles) in 2D, but also generalized Hopf indices in 3D, second Euler class in 4D, etc. I then introduce a systematic framework, based on the Pluecker imbedding of Grassmannians, for the modeling of crystalline phases of a given homotopy class. I will also show that this is a convenient starting point for studying of the Riemannian geometry associated with these topological phases, which naturally extends to many-band contexts. In the end, I will review a few examples of real material candidates and synthetic systems exhibiting non-Abelian (nodal) topologies and Euler topological phases.