The Haldane phase represents one of the most important symmetry-protected states in modern physics. This state can be realized using spin-1 and $\mathrm{spin}-\frac{1}{2}$ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond ${\mathbb{Z}}_3$ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as fourfold degeneracy in the ground-state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics in which braiding of the two edge modes yields a $\frac{2\pi }{3}$ phase. This model also supports rich phases, including a topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase, and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge $c=4/5$. Even in the topological FP phase, it is also characterized by long-range string order; thus we observe a drop of string order across the phase boundary between the FP phase and the Haldane phase. This work opens a new way for finding of exotic topological phases with parafermions.

• Received 25 March 2019
• Revised 16 September 2019

DOI:https://doi.org/10.1103/PhysRevB.100.165102