Duality analysis on random planar lattices

The conventional duality analysis is employed to identify a location of a critical point on a uniform lattice without any disorder in its structure. In the present study, we deal with the random planar lattice, which consists of the randomized structure based on the square lattice. We introduce the uniformly random modification by the bond dilution and contraction on a part of the unit square. The random planar lattice includes the triangular and hexagonal lattices in extreme cases of a parameter to control the structure. A modern duality analysis fashion with real-space renormalization is found to be available for estimating the location of the critical points with a wide range of the randomness parameter. As a simple test bed, we demonstrate that our method indeed gives several critical points for the cases of the Ising and Potts models and the bond-percolation thresholds on the random planar lattice. Our method leads to not only such an extension of the duality analyses on the classical statistical mechanics but also a fascinating result associated with optimal error thresholds for a class of quantum error correction code, the surface code on the random planar lattice, which is known as a skillful technique to protect the quantum state.

Figure 1

(top left) The unit cluster for the square lattice with four bonds $s=1$. (bottom left) The single bond once we consider the simple application of the conventional duality with the replica method. (right) The larger cluster with 16 bonds, $s=2$, used to identify a more precise location of the critical points. The black circles denote the spin variables that we sum over. The white circles express the spin variables without summation.

Figure 2

Structure of the random planar lattice. The bottom panels depict a part of the unit cell of the whole lattice. The two types of modification, “bond dilution” and “bond contraction,” are assigned on each dashed bond. The right panels give the dual lattice, while the left ones express the original lattice. Readers can confirm that the self-duality holds on the structure of the lattice even with the existence of a part of the modifications.

Figure 3

(left) One of the realizations of a random planar lattice and (right) its dual one.

Figure 4

Extreme limit of $p_{\mathrm{mix}.}=0$. The original lattice in the left panel becomes the triangular lattice. On the other hand, the dual one in the right panel is the hexagonal lattice. To guide the eye, we rearrange the location of the sites as the modifications of the unit cell in the bottom panels.

Figure 5

Extreme limit of $p_{\mathrm{mix}.}=1$. The original lattice in the left panels turns into the hexagonal lattice, while the dual one in the right panels becomes the triangular lattice. The same symbols as those in Fig. 4 are used.

Figure 6

Other random planar lattices where one can perform the duality analysis. The left panel shows a simple case by tiling of the unit cell with four bonds. The right panel shows a more complicated case with a unit cell with 16 bonds.

Figure 7

A random planar lattice where one cannot obtain the critical points. In the bottom left panel, we describe the unit cell. The extreme limit $p_{\mathrm{mix}}=0$ gives the $(4,8^2)$ lattices. This is the notation of the uniform tilings, in which all polygons are regular and each vertex is surrounded by the same sequence of polygons. The notation $(4,8^2)$ expresses that every vertex is surrounded by one square and two octagons.

Figure 8

The dual lattice of the random planar lattice depicted in Fig. 7. The same symbols as those in Fig. 7 are used. The extreme limit $p_{\mathrm{mix}}=1$ becomes the Union Jack lattice, which is the mutually dual pair of the $(4,8^2)$ lattice.

Figure 9

(left) The unit cluster for the random planar lattice $s=1$. (right) The larger cluster $s=2$ to estimate a more precise location for the critical points. The same symbols as in Fig. 1 are used.

Figure 10

(a) A qubit, denoted by an open circle, is located on each edge of the lattice. The stabilizer operators are defined on each face and vertex. (b) $Z$ error chains $E$ and $\tilde{E}$, which have the same error syndrome. The incorrect error syndromes are displayed by the gray shading. (c) The associated RBIM with the $Z$ error correction problem. An Ising spin is located on each vertex of the lattice. A coupling constant $J_{ij}^Z=f_{ij}\left(E\right)$ is assigned on each edge. The actual and hypothetical error chains $E$ and $\tilde{E}$ correspond to the antiferromagnetic interactions shown by the double lines. The excited domain wall is shown by the bold line. The gray shaded area is a domain of ${\sigma }_i=-1$. The crosses indicate frustrations. Equivalently, they are also the end points of the excited domain wall, the so-called Ising vortices. (d) Primal (solid lines and solid circles) and dual (dotted lines and open circles) lattices. According to the definition of the stabilizer operators shown in (a), the $Z$ and $X$ error correction problems are mapped to the RBIMs on the primal and dual lattices, whose concentrations of antiferromagnetic interactions are given by $p_Z$ and $p_X$, respectively.