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The $\nu = 2/3$ filling is the simplest paradigmatic example of a fractionalquantum Hall state, which contains counter-propagating edge modes. These modescan be either in the unequilibrated regime or equilibrated to differentextents, on top of the possible edge reconstruction. In the unequilibratedregime, two distinct renormalization group fixed points have been previouslyproposed, namely Kane-Fischer-Polchinski and Wang-Meir-Gefen. In theequilibration regime, different degree of thermal equilibration can exist,while charge is fully equilibrated. Here, we show that these rich variety ofmodels can give rise to three possible conductance plateaus at $e^2/2h$(recently observed in experiments), $5e^2/9h$ (predicted here), and $e^2/3h$(observed earlier in experiments) in a quantum point contact geometry. Weidentify different mechanisms for \emph{electrical shot noise} generation atthese plateaus, which provides an experimentally accessible venue fordistinguishing among the distinct models.

Shot noise classification of different conductance plateaus in a quantum point contact at the $\nu=2/3$ edge