Parafermions in hierarchical fractional quantum Hall states

Motivated by recent theoretical progress demonstrating the existence of non-Abelian parafermion zero modes in domain walls on interfaces between two dimensional Abelian topological phases of matter, we investigate the properties of gapped interfaces of hierarchical fractional quantum Hall states, in the lowest Landau level, characterized by the Hall conductance $\sigma_{xy}(m,p) = \frac{p}{2mp+1} \frac{e^{2}}{h}$, for integer numbers $(m,p)$ with $m,p \geq 1$. The case $m=1$ corresponds to the experimentally well established sequence of fractional quantum Hall states with $\sigma_{xy} = \frac{1}{3}\frac{e^{2}}{h}, \frac{2}{5}\frac{e^{2}}{h}, \frac{3}{7}\frac{e^{2}}{h}, ...\, $, which has been observed in many two dimensional electron gases. Exploring the mechanism by which the $(m,p+1)$ hierarchical state is generated from the condensation of quasiparticles of the ``parent" state $(m,p)$, we uncover a remarkably rich sequence of parafermions in hierarchical interfaces whose quantum dimension $d_{m,p}$ depends both upon the total quantum dimension $\mathcal{D}_{m,p} = \sqrt{2mp+1}$ of the bulk Abelian phase, as well as on the parity of the ``hierarchy level" $p$, which we associate with the $\mathbb{Z}_2$ stability of Majorana zero modes in one dimensional topological superconductors. We show that these parafermions reside on domain walls separating segments of the interface where the low energy modes are gapped by two distinct mechanisms: (1) a charge neutral backscattering process or (2) an interaction that breaks $U(1)$ charge conservation symmetry and stabilizes a condensate whose charge depends on $p$. Remarkably, this charge condensate corresponds to a clustering of quasiparticles of fractional charge $\frac{p}{2mp+1}\,e$, allowing us to draw a correspondence between these fractionalized condensates and Read-Rezayi non-Abelian fractional quantum Hall cluster states.