Quasiparticle agglomerates in Read-Rezayi and anti Read-Rezayi state

We calculate the dominant excitations for the $k$-level ($k\in\mathbb{N}$) Read-Rezayi (RR) states and their particle-hole conjugates, the anti Read-Rezayi ($\bar{\textrm{RR}}$), proposed for quantum Hall states. These states are supposed to be build over the second Landau level with total filling factor $\nu=2+\nu^*$ with $\nu^*=k/(k+2)$ for RR and $\nu^*=2/(k+2)$ for $\bar{\textrm{RR}}$. In the $k$-level RR states, based on $\mathbb{Z}_k$ parafermions, the dominant excitations are the fundamental quasiparticles with fractional charge $e^*_k= e/(k + 2)$, with $e$ the electron charge, if $k=2,3$. For k=4 the single-qp and the 2-agglomerate, with charge $2e^*_k$, have the same scaling and both dominate, while for $k>4$ the 2-agglomerates are dominant. Anyway the dominance of the 2-agglomerates can be affected by the presence of environmental renormalizations. For all the $k$-level $\bar{\textrm{RR}}$ states the single-qp and the 2-agglomerate have the same scaling and both dominate. In this case only the presence of environmental renormalizations can make one dominant over the other. We determine the conditions where the environmental renormalizations of the charged and neutral modes make dominant the Abelian 2-agglomerates over the non-Abelian single-quasiparticles in the two models and for any value of $k$. We conclude observing that, according these predictions, the dominance of 2-agglomerates, at very low energies for the $\nu=5/2$, can be an interesting indication supporting the validity of the anti-Pfaffian model in comparison to the Pfaffian.