Abstract
The stability of the topological order phase induced by the ${\mathbb{Z}}_{3}$ Kitaev model, which is a candidate for fault-tolerant quantum computation, against the local order phase induced by the three-state Potts model is studied. We show that the low-energy sector of the Kitaev-Potts model is mapped to the Potts model in the presence of transverse magnetic field. Our study relies on two high-order series expansions based on continuous unitary transformations in the limits of small and large Potts couplings as well as mean-field approximation. Our analysis reveals that the topological phase of the ${\mathbb{Z}}_{3}$ Kitaev model breaks down to the Potts model through a first-order phase transition. We capture the phase transition by analysis of the ground-state energy, one-quasiparticle gap, and geometric measure of entanglement.
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