.Ordering topological orders in quantum Hall systems and beyond

I will discuss possible topological orders/topological quantum field theories arising in quantum Hall systems. Given the value of the Hall conductivity, we constrain the anyon content of the topological order, using flux-threading argument and anomaly matching. These constraints are powerful enough to single out a unique minimal topological order (or, in some cases, a small set of minimal candidates), providing a natural organizing principle for the possible topological phases realized in such systems. Remarkably, almost all experimentally observed quantum Hall topological orders correspond to these minimal theories.

I will then discuss applications of these ideas in lattice systems of fermions with a given filling factor. Owing to the filling anomaly, insulating states that preserve lattice translation symmetries -- "quantum charge liquids" -- must either be topologically ordered, or host gapless neutral excitations. We prove several results characterizing the minimal topological orders consistent with a given lattice filling. In particular we show that, at rational fillings 𝜈=𝑝/𝑞 with 𝑞 an even integer, the minimal ground-state degeneracy on a torus of the gapped quantum charge liquid is 4⁢𝑞^2, four times larger than that of the bosonic QCL at the same filling.