Determination of topological edge quantum numbers of fractional quantum Hall phases by thermal conductance measurements.

To determine the topological quantum numbers of fractional quantum Hall (FQH) states hosting counter-propagating (CP) downstream (N d ) and upstream (N u ) edge modes, it is pivotal to study quantized transport both in the presence and absence of edge mode equilibration. While reaching the non-equilibrated regime is challenging for charge transport, we target here the thermal Hall conductance G Q , which is purely governed by edge quantum numbers N d and N u . Our experimental setup is realized with a hexagonal boron nitride (hBN) encapsulated graphite gated single layer graphene device. For temperatures up to 35 mK, our measured G Q at ν = 2/3 and 3/5 (with CP modes) match the quantized values of non-equilibrated regime (N d  + N u )κ 0 T, where κ 0 T is a quanta of G Q . With increasing temperature, G Q decreases and eventually takes the value of the equilibrated regime ∣N d  - N u ∣κ 0 T. By contrast, at ν = 1/3 and 2/5 (without CP modes), G Q remains robustly quantized at N d κ 0 T independent of the temperature. Thus, measuring the quantized values of G Q  in two regimes, we determine the edge quantum numbers, which opens a new route for finding the topological order of exotic non-Abelian FQH states.