Quantum phase transitions out of a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological phase

We investigate the low-energy spectral properties and robustness of the topological phase of color code, which is a quantum spin model for the aim of fault-tolerant quantum computation, in the presence of a uniform magnetic field or Ising interactions, using high-order series expansion and exact diagonalization. In a uniform magnetic field, we find first-order phase transitions in all field directions. In contrast, our results for the Ising interactions unveil that for strong enough Ising couplings, the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological phase of color code breaks down to symmetry broken phases by first- or second-order phase transitions.

Figure 1

TCC on the honeycomb lattice $\Lambda$ placed on a torus with genus $g=1$. The coding subspace of the system is spanned by string operators. A blue plaquette $p_1$ is characterized by two red and green closed strings at its boundary. An open red string (${\mathcal{C}}_{\gamma }$) creates two $X$ particles on the surface of the two red plaquettes $p_2$ and $p_3$. For every homology class of the torus, there are four global strings which can be labeled as (${\mathcal{C}}_1$, $...$, ${\mathcal{C}}_4$). The global strings are responsible for the topological degeneracy 16 of the ground state.

Figure 2

Honeycomb clusters with periodic boundary condition. Closed circles are the physical spins at the vertices of the lattice and the dashed circles denote the periodicity at the boundaries of the lattice. The dotted gray lines further demonstrate the two site unit cells which span the lattice.

Figure 3

TCC on the dual triangular lattice. The colored vertices represent the plaquettes of the same color on the original honeycomb lattice. A single parallel Ising perturbation is mapped to three Ising interactions on the nearest-neighbor sites $p,p^{\prime }$ of the colored triangular sublattices formed by connecting the vertices of the same color.

Figure 4

Phase diagram of the TCC perturbed by parallel perturbations $(j_x,j_z)$ or $(h_x,h_z)$ setting $J=1$. The boundaries of the yellow region correspond to second-order phase transitions of the TCC in the presence of Ising interactions $(j_x,j_z)$ obtained from the 1-QP gap in the small-coupling limit. The circles are the corresponding phase boundaries obtained by ED on 18- and 24-site periodic clusters. The red dashed line and squares represent first-order phase transitions calculated by ED on periodic 18 and 24-site clusters. The red diamonds on the axis and on the isotropic line ($h_x=h_z$) are the corresponding first-order points obtained by pCUT.[33] The inset further shows how the increase in system size sharpens the resonances of the $-{\partial }_j^2{\varepsilon }_0$ for $j_x=j_z=j$ in ED calculations.

Figure 5

Transition point $j_c$ (a) and the critical exponent $z\nu$ (b) of the triangular TFIM extrapolated obtained from DlogPadé approximants as a function of the order $r$ of the series for $J=1$. Here $L$ is given by $L=$[$r-(r$mod$2)$]$/2$. The literature values $j_c=0.209$ and $z\nu =0.630$[35, 36] are highlighted by dashed lines.

Figure 6

Transition point $j_c$ ($J=1$) and critical exponent $z\nu$ of the TCC at the multicritical point ($j_x=j_z,j_y=0$) extracted from the DlogPadé approximants [$L,L$], [$L+1,L$] and [$L,L-1$] with $L=$[$r-(r$mod$2)$]$/2$ of the one-particle gap $\Delta$ obtained by pCUT.

Figure 7

(a) A two-particle bound state with even particle number parity. The two bounded particles hop to the neighboring blue plaquette in order two perturbation theory. (b) A three-particle bound state with odd parity on the plaquettes. The three bounded $X$-type QPs turn together into $Z$-type particles (or vice versa) in odd orders of perturbation.

Figure 8

The ground-state energy per spin ${\varepsilon }_0$ of the TCC plus transverse perturbations $j_y$ or $h_y$ as a function of $\theta$ in units of $\pi /2$ setting $J=\cos \theta$ and $j_y,h_y=\sin \theta$. The discontinuities in ${\varepsilon }_0$ signal first-order phase transitions at ${\theta }_c^j\approx 0.62$ for $j_y$ and at ${\theta }_c^h\approx 0.84$ for $h_y$ (indicated by vertical dashed lines).

Figure 9

Low- and high-field gap $\Delta$ of the TCC in transverse magnetic field as a function of $\theta$ such that $J=\cos \theta$ and $h_y=\sin \theta$. Solid lines correspond to the bare series of maximum order 8, 10, and 9 for 1-, 2- and 3-QP low-field gaps, respectively, and order 16 for high-field 1-QP gap. The symbols also denote different Padé approximants. The vertical gray dashed line represents the transition point obtained by analyzing ground-state energies (see Fig. 8).

Figure 10

Phase diagram of the TCC plus Ising interactions ($j,j_y$) with $j\equiv j_x=j_z$ and $J=1$. Solid lines are obtained by pCUTs while red and green symbols correspond to ED data of periodic 18- and 24-site clusters, respectively. The blue diamond represents the transition point obtained by pCUT for ($0,j_y$). The yellow region illustrates the topologically ordered phase while the cyan/gray shaded area represents a plane of first-order phase transitions. Inset: Critical exponent $z\nu$ as a function of $j_y$ obtained from different DlogPadé extrapolants $[L,M]$ of the 1-QP gap.

Figure 11

The 1-QP gap of the perturbed TCC on the isotropic line ($j\equiv j_x=j_y=j_z$) for different DlogPadé approximants. Here we have set $J=1$. The upper inset depicts $-{\partial }_j^2{\varepsilon }_0$ obtained from ED for periodic clusters with 18 and 24 sites. The critical point highlighted by gray dashed line is consistent with the closure of the gap at $j_c\approx 0.25$. The lower inset displays the first-order transition points on the isotropic line which merge into the second-order point obtained by the closure of the gap.