Emergent Haldane phase in an alternating bond $\mathbb{Z}_3$ parafermion chain

TL;DR

This paper demonstrates the emergence of a Haldane phase in an alternating bond $ ext{Z}_3$ parafermion chain, revealing novel topological properties, symmetry protections, and phase transitions in parafermionic systems.

Contribution

It introduces a new realization of the Haldane phase in a $ ext{Z}_3$ parafermion chain, highlighting its topological features and phase diagram, distinct from previous models.

Findings

Emergent Haldane phase characterized by long-range string order.

Ground state degeneracy and entanglement spectra reveal topological protection.

Rich phase diagram including topological, trivial, dimer, and gapless phases.

Abstract

The Haldane phase represents one of the most important symmetry protected states in modern physics. This state can be realized using spin-1 and spin-${1\over 2}$ 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 four-fold 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 ${2\pi \over 3}$ phase. This model also supports rich phases, including 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 the long-range string order, thus we observe a drop of string order across the phase boundary between the FP phase and Haldane phase. These phenomena are quite general and this work opens a new way for finding exotic topological phases in $\mathbb{Z}_k$ parafermion models.