Robustness of a topological phase: Topological color code in parallel magnetic field

TL;DR

This paper investigates the stability of the topological color code under a uniform magnetic field, revealing its robustness and phase transition characteristics through various analytical and numerical methods.

Contribution

It demonstrates that the topological color code remains stable up to a critical field and is more robust than the toric code, using high-order series expansions and exact diagonalization.

Findings

Topological phase stable up to a critical magnetic field

Phase transition is first-order to a polarized phase

Color code more robust than the toric code in parallel magnetic field

Abstract

The robustness of the topological color code, which is a class of error correcting quantum codes, is investigated under the influence of an uniform magnetic field on the honeycomb lattice. Our study relies on two high-order series expansions using perturbative continuous unitary transformations in the limit of low and high fields, exact diagonalization and a classical approximation. We show that the topological color code in a single parallel field is isospectral to the Baxter-Wu model in a transverse field on the triangular lattice. It is found that the topological phase is stable up to a critical field beyond which it breaks down to the polarized phase by a first-order phase transition. The results also suggest that the topological color code is more robust than the toric code, in the parallel magnetic field.