Noninteracting tight-binding models for Fock parafermions

TL;DR

This work shows how to construct noninteracting tight-binding models for four-state Fock parafermions in one dimension by mapping to spin-1/2 fermions. The single-particle spectrum is real and identical to that of an underlying fermionic model, allowing the many-body energies to be written as $E=\sum_{\ell=1}^L n_{\ell}\epsilon_{\ell}$ with $n_{\ell}\in\{0,1,2,3\}$. The authors illustrate with Rice-Mele and SSH-like chains and verify a thermodynamic occupation function that matches the mapped fermion picture, including a down-spin branch at effective temperature $T/2$. They further construct a parafermionic Kitaev chain whose topological phase hosts a fourfold ground-state degeneracy associated with Majorana edge modes, highlighting potential experimental realizations and extensions to higher-$p$ parafermions.

Abstract

By mapping itinerant spin-$1/2$ fermions to four-state Fock parafermions, we construct noninteracting tight-binding models for Fock parafermions in one dimension. They have single-particle real energy spectra consisting of a sum of single-particle energy levels each multiplied by a parafermionic occupation number. The single-particle levels may be determined by diagonalizing a square matrix whose order scales linearly with system size. These levels are the same as those of noninteracting fermionic models, as we show explicitly for the Rice-Mele model and the Su-Schrieffer-Heeger model. We show that the thermodynamic distribution function for the occupation numbers of noninteracting four-state parafermions is consistent with the mapping to spin-$1/2$ fermions. We apply the mapping to create a parafermionic counterpart of the Kitaev superconducting chain and show that the ground state in the topological phase is fourfold degenerate, with each ground state distinguished by the fourfold parafermionic occupation numbers of Majorana edge modes.