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

The robustness of the topological color code, which is a class of error-correcting quantum codes, is investigated under the influence of a 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.

Figure 1

A piece of 2-colex on a honeycomb lattice placed on a torus. Colored circles at the vertices of the lattice represent spins $\frac{1}{2}$. A colored plaquette such as $p_1$ is characterized by two closed strings of the other two colors. $X_1,...,X_4$ are the global strings of the model on a periodic structure such as a torus with $g=1$. The elementary excitations are described by open strings such as $\gamma$ which create particles at their end points on the surface of a plaquette (see $p_2$) which share the same color with the string.

Figure 2

Every time a ${\sigma }_{\mathfrak{v}}^x$ acts on the site $\mathfrak{v}$ of the lattice, it can be considered as the endpoint of three different colored open strings say (red, green, blue) which create three $Z$-type quasiparticle (QP) on the neighboring plaquettes sharing the same color with the strings. The yellow circle denotes the ${\sigma }^x$ field which is connected to the endpoints of the red, blue, and green strings and the three colored circles on the surface of the plaquettes are the QP excitations in the topological phase. The QPs are created on the vertices of the effective triangular lattice.

Figure 3

Ground-state energy per site ${\epsilon }_0^{\mathrm{lf}}$ (upper panel) and the single-particle gap ${\Delta }^{\mathrm{lf}}$ (lower panel) as a function of $h_x/2J$ in the low-field limit. Solid lines (symbols) represent bare series (Padé and dlog Padé approximants). The coupling range between the dashed vertical lines corresponds to the location of the first-order phase transition deduced from the crossing points of the ground-state energy given in Table .

Figure 4

Ground-state energy per site ${\epsilon }_0^{\mathrm{hf}}$ (upper panel) and the six-particle gap ${\Delta }^{\mathrm{hf}}$ (lower panel) as a function of $J/2h_x$ in the high-field limit. Solid lines (symbols) represent bare series (Padé and dlog Padé approximants). The coupling range between the dashed vertical lines corresponds to the location of the first-order phase transition deduced from the crossing points of the ground-state energy given in Table .

Figure 5

The ground-state energy per site ${\epsilon }_0$ as a function of $\theta$. Solid line corresponds to the bare series of order 10. Circles denote different Padé approximants and the squares represent the exact diagonalization results for different lattice sizes.

Figure 6

The magnetization $M$ of the system as a function of $\theta$. Solid line corresponds to the bare series of order 10. Symbols denote different Padé approximants.

Figure 7

The ground-state susceptibility $\chi$ of the system as a function of $\theta$. Solid line corresponds to the bare series of order 10. Symbols denote different Padé approximants.

Figure 8

Low- and high-field gap $\Delta$ as a function of $\theta$ for different dlog Padé approximants.