Fermionizing Parafermions

Parafermion zero modes are Majorana-fermion generalizations that exhibit comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in 2D fractionalized media, to strictly 1D fermionic systems. Most notably, Z₄ parafermions map onto physical electrons enjoying time-reversal symmetry. Phases for the former translate into a variety of interesting electronic states, including a time-reversal-invariant topological superconductor (TRITOPS) hosting Kramers pairs of edge Majorana zero modes. We further show that nontrivial parafermionic fusion rules survive in their electronic counterparts, and can be detected via a novel pumping cycle for the TRITOPS phase that yields an edge magnetization with quadrupled periodicity. We generalize these results to arbitrary Z_2N parafermion chains. Our work highlights new avenues for exploring 'beyond-Majorana' physics in experimentally realistic, weakly interacting 1D electronic platforms.