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.