Non-Abelian fermionization and the landscape of quantum Hall phases

The recent proposal of non-Abelian boson-fermion dualities in $2+1$ dimensions, which morally relate $U{\left(k\right)}_N$ to $SU{\left(N\right)}_{-k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian “composite fermions” partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian, $U{\left(k\right)}_2$ Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across $N$ layers can also generate the family of Blok-Wen states with $U{\left(k\right)}_2$ topological order.

Figure 1

Proposed phase diagram for the $SU\left(2\right)$ composite fermion theory at filling $\nu =3/2$. Here $\overline{\lambda }=g_{YM}^2/{\omega }_c$, where $g_{YM}^2$ is the Yang-Mills coupling (the fine structure constant) and ${\omega }_c\sim \sqrt{B}$ is the cyclotron frequency. When $\overline{\lambda }\to \infty$, the Yang-Mills term vanishes, and the picture of deconfined composite fermions filling (color degenerate) Landau levels is valid, leading to the $SU{\left(2\right)}_{-3}^{\mathrm{spin}}\leftrightarrow U{\left(3\right)}_2$ state. When $\overline{\lambda }$ runs small, the Yang-Mills term becomes very large, Landau levels can mix, and the deconfinement of the composite fermions is no longer assured, leading ultimately to the $U{\left(1\right)}_2$ phase predicted in the dual Abelian theory. These states are separated by a critical point at $\overline{\lambda }={\overline{\lambda }}_{*}\sim O\left(1\right)$, which is likely first order.