UV-IR transmutation for hybrid realizations of Z_k parafermion systems

Recent experiments brought Majorana particles closer to reality and raised interest in realizing $\mathbb{Z}_k$ parafermions for general value of $k$ in hybrid structures. Of particular interest is the prospect of realizing a two dimensional system with the edge states described by $\mathbb{Z}_k$ parafermion conformal field theory (CFT) by coupling one dimensional hybrid systems. However in order to understand the effective perturbation due to local inter-wire coupling, it is crucial to relate short distance ultra-violet (UV) degrees of freedom to primary fields of the CFT describing long distance infra-red (IR) physics. Here we consider two recent proposals for realizing a one dimensional system described by $\mathbb{Z}_k$ parafermion CFT upon tuning to quantum critical points: a parafermion chain and a pair of $\nu=2/k$ quantum Hall chiral edge states under uniform backscattering and pairing, where the latter is represented by self-dual sine-Gordon model. We explicitly derive the relation between distinct UV degrees of freedom relevant for each proposals and the primary fields of $\mathbb{Z}_3$ parafermion CFT. Our results point to a marked difference in the effect of local coupling between wires represented by each proposals, regarding robustness of the edge states in the resulting two dimensional systems.