A wave-function approach to the Kramers-Wannier self-duality in critical quantum chains

The Kramers-Wannier self-duality at critical points in classical statistical or quantum many-body systems provides one prototypical manifestation of non-invertible symmetries. In this talk, we reexamine the Kramers-Wannier self-duality in lattice systems from the perspective of model wave functions, taking the critical quantum Ising spin chain as a concrete example. We demonstrate that the symmetry operator for the Kramers-Wannier self-duality follows in a simple and direct way from a 'half-step' translation symmetry of the model wave function in the anyonic fusion basis. This translation operator, in turn, is expressed in terms of certain F-symbols in the fusion category of topological lines in the CFT describing the scaling limit of the model.