Bulk-measurement-induced boundary phase transition in toric code and gauge Higgs model

Study of boundary phase transition in toric code under cylinder geometry via bulk projective measurement is reported. As the frequency of local measurement for bulk qubits is increased, spin-glass type long-range order on the boundaries emerges indicating spontaneous-symmetry breaking (SSB) of $Z_2$ symmetry. From the lattice gauge theory viewpoint, this SSB is a signal of a transition to Higgs phase with symmetry-protected topological order. We numerically elucidate the properties of this phase transition in detail, especially its criticality, and give a physical picture using nonlocal gauge-invariant symmetry operators. Phase transition in the bulk is also studied and its relationship to the boundary transition is discussed.

Figure 1

Schematic figure of cylinder system with rough boundaries. This figure displays only the upper rough boundary. The star operator (the red-shaded object) and plaquette operator (right-blue shaded object) are considered. The red dashed regime is an upper rough boundary regime.

Figure 2

Schematic figure of an extended lattice gauge Higgs model with rough boundaries. The lattice in the $x$ direction is periodic and in $y$ direction is open. The red-dashed lines indicate the regime upper and lower boundaries.

Figure 3

Schematic image of phase diagram of the Hamiltonian $H_{\mathrm{TC}}$ on the cylinder geometry. We assume $h_x$ and $h_z$ are some constant. The dashed arrow represents our target measurement protocol related to the parameter sweep in the Hamiltonian formalism.

Figure 4

Behavior of the average value of ${\chi }_{SG}/L_x$. The right-blue shaded line represents the crossing point of the different system-size data. We used $3\times {10}^3$ samples. We fix $L_y=6$.

Figure 5

(a) System-size dependence of $F_v$. (b) Scaling function. (c) Correlation for $C_{SG}({\ell }_x,{\ell }_x^{\prime })$. Here, we chose ${\ell }_x=0, L_x=72$. We used the following fitting functions: $f_1\left({\ell }_x^{\prime }\right)=aexp(-{\ell }_x^{\prime }/\xi )+b$ and $f_2\left({\ell }_x^{\prime }\right)=a{\ell }_x^{\prime \eta }+b$, where $a, b, \xi$, and $\eta$ are fitting parameters. Scatter plots represent numerical results specified by probability $p$. Note that peak of $F_v$ in $L_x=72$ system is located at $p=0.66$. Thus, physical quantities are expected to show critical behavior at $p=0.66$ in $L_x=72$ system. We used $(3-4)\times {10}^3$ samples for (a)–(c).

Figure 6

Partition for $S_{AF}$ (a) and $S_{AH}$ (b). Sample-to-sample variance for $S_{AF}$ (c) and $S_{AH}$ (d). (e) System-size dependence of $S_{AH}$, where we averaged over all measurement samples. We used $3\times {10}^3$ samples for (c), (d), and (e).

Figure 7

System-size dependence of $S_{AF}$, where we averaged over all measurement samples. We used $3\times {10}^3$ samples. We fix $L_y=6$.

Figure 8

Histogram of the number of ${\sigma }_x$ in each generator of stabilizers for $0.3\le p\le 0.9$. We omit the data point for zero number of ${\sigma }^x$ and $S_x^u$. We used $3\times {10}^3$ samples.

Figure 9

$L_y$ dependence of spin-glass order ${\chi }_{SG}/L_x$. (a) $L_y=2$, (b) $L_y=3$, and (c) $L_y=4$. Each inset panel represents $F_v$. We used $4\times {10}^3$ samples.

Figure 10

The sample-to-sample variance of $\overline{A}$. The inset displays the $L·\overline{A}$. We used $3\times {10}^3$ samples. We fix $L_y=6$.