Non-abelian bosonization and the quantum Hall states at $ u=5/2$

The quantum Hall plateau at the filling fraction $ u=5/2$ has garnered attention due to the proposal that it hosts non-abelian anyons. Historically, there have been two major candidates for the quantum Hall state at $ u=5/2$ – the Pfaffian state and the anti-Pfaffian state. A more recent series of theoretical works, to match the theory with thermal Hall conductance experimentally measured for the $ u=5/2$ plateau, proposed a picture akin to the Chalker-Coddington network model in which the quantum Hall state at $ u=5/2$ are composed of puddles of Pfaffian and anti-Pfaffian states induced by disorder. In this work, we apply field-theoretical methods to study the Pfaffian-anti-Pfaffian network and show how the strong-disorder fixed point obtained from non-abelian bosonization can be employed to access both the insulating phase of the network consistent with the quantum Hall plateau at $ u=5/2$ and the thermal metal phase. While these two phases were previously known to appear in the network of Pfaffian and anti-Pfaffian states, non-abelian bosonization provides a unified framework to understand how the two different phases can emerge in the same network. We support our analytical insight with numerical simulations as well.