Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.
A simple quantum chain with a topological phase is constructed. It is formed by parafermions (quasiparticles obeying the fractional statistics) with angular parameter
θ=2π/2π33 . The model is derived from the one-dimensional spin model with topological order protected by the
Z3×Z3 symmetry (generalized cluster model). Parafermionic zero modes arising at the chain boundaries, which remain stable because of the vacuum gap and topological order, are constructed and studied.
Nonlocal (topological) order and local (Landau-type) order are often viewed as opposing principles. Here, the authors study strongly interacting quantum wires that exhibit hybrid order, composed of local Ising magnetization and a topological order parameter. Based on an exact mapping of the corresponding strongly interacting electronic Hamiltonian onto \ensuremath{\mathbb{Z}}
n parafermions, they show that the system can be in a hybrid phase which is distinct from the topological and from the Ising phase. Moreover, the system has remarkable similarities to superconducting hybrid devices built from fractional quantum Hall edge states, which can host parafermions useful for quantum computation.