Parafermionic representation of Potts-based cluster chain

The cluster chain with $\mathbb{Z}_p \times \mathbb{Z}_p$ symmetry-protected topological (SPT) order is decomposed into two distinct bilinear parafermionic chains, each possessing intrinsic topological order. These chains are formed by standard parafermions and time-reversal parafermions, respectively. Each subsystem retains its own $\mathbb{Z}_p$ symmetry component, which characterizes the total parity of constituent particles. Their topological orders are inherited from the two SPT orders of the cluster model. The transformations of particles under reflection, translation, and time reversal are derived. In the open chain, four zero-energy parafermionic edge modes are identified, and their structure is analyzed. For the closed system, the boundaries are twisted by the total parafermion parity. It is shown that the open chain is reflection-invariant when the number of spins is even, and $\mathcal{PT}$-invariant when the number of spins is odd. Meanwhile, the closed model exhibits a symmetry characterized by an antiunitary analog of the dihedral group.