Anti-topological crystal and non-Abelian liquid in twisted semiconductor bilayers

We show that electron crystals compete closely with non-Abelian fractional Chern insulators in the half-filled second moiré band of twisted bilayer MoTe₂. Depending on the twist angle and microscopic model, these crystals can have non-zero or zero Chern numbers C. The C = 0 crystal occurs because contributions to the total Chern number from the full first band (+1) and half-full second band (-1) cancel. This is counterintuitive because the first two non-interacting bands in a given valley have the same Chern number  + 1. For these two reasons, we call this crystal an anti-topological crystal. The anti-topological crystal is a novel type of electron crystal that may occur in systems with multiple Chern bands at filling factors n > 1.