CMT Forum: Yang Bo

Conformal Hilbert Spaces in Chern Bands and non-Abelian anyons in Abelian topological phases

Strongly interacting electrons confined to a two-dimensional Chern band lead to collective anyonic excitations that can potentially be manipulated for the robust storage and processing of quantum information. These anyons, however, are not featureless point particles; under realistic conditions they have rich dynamics from inter-anyon interaction, and we can even probe the internal structures of individual anyons. Using the Landau levels as the simplest example, we introduce the formalism of conformal Hilbert spaces as the theoretical tool for characterizing the algebraic structure of the many-body Hilbert spaces in Chern bands. The formalism is particularly useful for characterising the nature of and the relationship between different types of anyons. Among several applications we will focus on the possibility of accessing non-Abelian anyons in Abelian fractional topological phases, and the "high energy physics" of gapped excitations emerging from the interplay between quantum geometry and topology. We will also outline how this formalism can be generalized to moire Chern bands, and their experimental consequences.