Determination of Topological Edge Quantum Numbers of Fractional Quantum Hall Phases

The previous chapter was primarily focused on the thermal conductance \((G_Q)\) of the hole-conjugate fractional quantum Hall (FQH) states \((\nu =5/3, 8/3)\) . The measured values of \(G_Q= 3\kappa _{0}T\) and \(4\kappa _{0}T\) ( \(\kappa _{0}T\) is a quantum of \(G_{Q}\) ) for these states suggest the thermally non-equilibrated regime of heat transport for the 2/3 structure of the edge modes. This is true if we assume the widely accepted edge structure of 2/3 state harbouring one downstream and one upstream mode (MacDonald 1990; Johnson and MacDonald 1991; Wen 1995). However, in principle, a 2/3 edge with two co-propagating downstream modes (Beenakker 1990) is also a fully legitimate FQH edge, which will correspond to the same values of \(G_Q\) observed in the previous chapter. To resolve the dichotomy between these two models or, in general, 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 the thermal Hall conductance \(G_{Q}\) , governed by edge quantum numbers \(N_d\) and \(N_u\) . Our experimental setup is realized with an encapsulated graphite-gated monolayer graphene device. For temperatures up to 35 mK, our measured \(G_{Q}\) at \(\nu = \) 2/3 and 3/5 (with CP modes) match the quantized values of non-equilibrated regime \((N_d + N_u)\kappa _{0}T\) . With increasing temperature, \(G_{Q}\) decreases and eventually takes the value of equilibrated regime \(|N_d - N_u|\kappa _{0}T\) . By contrast, at \(\nu = \) 1/3 and 2/5 (without CP modes), \(G_Q\) remains robustly quantized at \(N_d\kappa _{0}T\) independent of the temperature. Thus, measuring the quantized values of \(G_{Q}\) at two regimes, we determine the edge quantum numbers, which opens a new route for finding the topological order of exotic non-Abelian FQH states.