A non-Abelian twist to integer quantum Hall states

Through a theoretical coupled wire model, we construct strongly correlated electronic \emph{integer} quantum Hall states. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law as $\left({\kappa }_{xy}\mathrm{/}{\sigma }_{xy}\right)\mathrm{/}\left[\left({\pi }^2{k}_B^{2}T\right)\mathrm{/}3e^2\right]<1$.We propose a new Abelian incompressible fluid at filling $\nu =16$ that supports a bosonic chiral $(E_8{)}_1$ conformal field theory at the edge and is intimately related to topological paramagnets in (3+1)D. We further show that this topological phase can be partitioned into two non-Abelian quantum Hall states at filling $\nu =8$, each carrying bosonic chiral $(G_2{)}_1$ or $(F_4{)}_1$ edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the $E_8$ state that relates the $G_2$ and $F_4$ Fibonacci states.