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Although numerical studies modeling the quantum hall effect at fillingfraction $5/2$ predict either the Pfaffian (Pf) or its particle hole conjugate,the anti-Pfaffian (aPf) state, recent experiments appear to favor a quantizedthermal hall conductivity with quantum number $K=5/2$ , rather than the value$K=7/2$ or $K=3/2$ expected for the Pf or aPF state, respectively. While aparticle hole symmetric topological order (the PH-Pfaffian) would be consistentwith the experiments, this state is believed to be energetically unfavorable ina homogenous system. Here we study the effects of disorder that are assumed tolocally nucleate domains of Pf and aPf. When the disorder is relatively weakand the size of domains is relatively large, we find that when the electricalHall conductance is on the quantized plateau with $\sigma_{xy} = (5/2)(e^2/h)$,the value of $K$ can be only 7/2 or 3/2, with a possible first-order-liketransition between them as the magnetic field is varied. However, forsufficiently strong disorder an intermediate state might appear, which weanalyze within a network model of the domain walls. Predominantly, we find athermal metal phase, where $K$ varies continuously and the longitudinal thermalconductivity is non-zero, while the electrical Hall conductivity remainsquantized at $(5/2)e^2/h$. However, in a restricted parameter range we find athermal insulator with $K=5/2$, a disorder stabilized phase which isadiabatically connected to the PH-Pfaffian. We discuss a possible scenario torationalize these special values of parameters.

Topological Order from Disorder and the Quantized Hall Thermal Metal: Possible Applications to the $\nu = 5/2$ State