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.