The Mode-Shell correspondence, a unifying phase space theory in topological physics

In quantum and classical wave systems, several properties are topologically protected. Due to their enhanced robustness, these properties have attracted significant interest in recent decades. The most studied historical case is the existence of unidirectional edge states in topological insulators, governed by the bulk-edge correspondence, which links their existence to a topological index defined in the material's bulk. Many distinct, yet similar, results have been obtained for Dirac/Weyl cone isolated in wavenumber in topological semimetals, corner modes in higher order insulators or modes localised both in position and wavenumber like in the quantum valley Hall effect.

In this talk, I will explain how all these examples can be understood through a unifying theory: the mode-shell correspondence. This formalism relates the existence of isolated topological modes in phase space (in position or/and wavenumber) to a topological invariant defined on a surface enclosing these modes, forming a shell. These invariants are a phase-space generalization of Chern and winding marker and reduces to Chern or winding numbers in the semiclassical limit.