Robustness of a topological phase: Topological color code in a 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.

Quantum phase transitions out of aZ2×Z2topological phase

“…For the TCC in a uniform magnetic field we find first-order transitions to a polarized phase for all field directions. In the presence of a single parallel field, the system is mapped to the BaxterWu model in transverse field and undergoes a first-order transition at h c x ≈ 0.383 33 . When the two parallel fields are turned on, the phase diagram of the model has a rectangular shape in which the phase boundaries originate from the Baxter-Wu limit on the axis and merge into a transition point at h x = h z ≈ 0.42 on the isotropic line.…”

“…the Hilbert space decouples into subspaces having fixed configurations of x p eigenvalues. The low-energy physics is always contained in the sector where x p = +1 for all p 33 . The TCC in a single field h x can be mapped to the Baxter-Wu model 34 in transverse magnetic field by a du- …”

Quantum robustness of fracton phases

“…Apart from understanding the role of thermal fluctuations on fracton codes, it is important and interesting to investigate the breakdown of fracton topological order at zero temperature. For the more conventional topological models like the toric code, color codes, or stringnet models very rich phase diagrams with quantum critical behavior have been found when adding perturbations like an external magnetic field [8,[53][54][55][56][57][58][59][60][61][62][63][64][65]. This is widely unexplored for three-dimensional fracton order, which is the main motivation for this work.…”

Topological Z2 resonating-valence-bond quantum spin liquid on the ruby lattice

“…Quantum spin liquids (QSL) [1] are intriguing phases of matter which emerge in different systems ranging from quantum antiferromagnets [2][3][4][5] to superconducting phases [6,7] and topologically ordered systems [8][9][10][11][12][13][14][15][16][17][18]. Due to strong zero-point quantum fluctuations and lack of conventional magnetic long-range order, they are very difficult to detect and their experimental realization is still a challenge [1].…”