Probing bilayer topological order with layer-resolved transport

Shot noise has been used to measure fractional charges of anyons. The value of the charge imposes constraints on fractional statistics but does not determine it. This issue is particularly important in multi-component systems. For example, the zero charge of neutral anyons in bilayer graphene gives no information about their statistics at all. We propose a protocol to probe the statistics of charged and neutral anyons in multi-component systems with layer-resolved or spin-resolved noise. The protocol applies to the fractional quantum spin Hall effect in MoTe$_2$, topological states in multi-layer graphene and bilayer GaAs, and to recently discovered fractional excitons in bilayer graphene. The approach relies on the relation between statistics and the distribution of the anyon charge over the components. Information about statistics can also be extracted from a simpler measurement of the layer-resolved electric current through a narrow constriction in a Hall bar even in the presence of long-range interactions and other non-universal effects.