Foliated fracton order in the checkerboard model

In this work, we show that the checkerboard model exhibits the phenomenon of foliated fracton order. We introduce a renormalization-group transformation for the model that utilizes toric code bilayers as an entanglement resource and show how to extend the model to general three-dimensional manifolds. Furthermore, we use universal properties distilled from the structure of fractional excitations and ground-state entanglement to characterize the foliated fracton phase and find that it is the same as two copies of the X-cube model. Indeed, we demonstrate that the checkerboard model can be transformed into two copies of the X-cube model via an adiabatic deformation.

Figure 1

(a) $A\text{−}B$ checkerboard bipartition of cubic lattice cells. The darkened cells belong to the $A$ sublattice. Black dots represent qubits. (b) $X_c$ and $Z_c$ Hamiltonian terms. Here, $\prod X$ ($\prod Z$) denotes a product of $X$ ($Z$) operators over the depicted qubits.

Figure 2

Qubits involved in the RG transformation for the checkerboard model. A single unit cell of the original $2L_x\times 2L_y\times 2L_z$ cubic lattice is depicted here. The black qubits belong to the original checkerboard model. The red and blue qubits comprise the toric code bilayer used as an entanglement resource in the RG procedure and are placed at the vertices of square lattices which are respectively embedded in the $z=a$ and $z=b$ planes. The shaded cube belongs to the $A$ sublattice of the checkerboard bipartition.

Figure 3

Action of the local unitary $S$ on the stabilizer generators of the composite ground state $|{\psi }_{\mathrm{CB}}\rangle \otimes |{\psi }_{\mathrm{TC}}^a\rangle \otimes |{\psi }_{\mathrm{TC}}^b\rangle$. Here $\prod X$ $(\prod Z)$ denotes the product of Pauli $X$ ($Z$) operators over all depicted qubits. On the left side, the shaded cells correspond to the original $A$ sublattice, whereas on the right side shaded cells correspond to the enlarged $A$ sublattice.

Figure 4

Modified checkerboard sublattice structure after the red and blue qubit layers have been incorporated into the model via the RG transformation. The new $A$ sublattice corresponds to the shaded cells.

Figure 5

An example of a checkerboard lattice structure embedded in $S^2\times S^1$. Depicted here is an $S^2$ cross section. The closely spaced adjacent circles represent bilayers, and the shaded cells belong to the $A$ sublattice.

Figure 6

(a) 3D solid torus $I(A;B|C)$ scheme and (b) 3D wire frame $I(A;B|C)$ scheme. In both cases the regions are contained within an overall cube of side length $L$.

Figure 7

Matching of qubits between the checkerboard model and two copies of the X-cube model tensored with ancilla qubits. A $2\times 2\times 2$ cell of the checkerboard model cubic lattice is shown here, corresponding to a single unit cell of $\mathrm{\Lambda }$, whose vertices lie at the green points. Shaded cubes belong to sublattice $A$ of the checkerboard bipartition. The red and blue qubits located respectively on the direct lattice (solid lines) and dual lattice edges (dashed lines) belong to the two X-cube copies, whereas the green and purple qubits at the vertices and body-center are ancilla degrees of freedom. The numbers label the qubits of a single unit cell of $\mathrm{\Lambda }$.

Figure 8

Examples of (a) a wire-frame operator and (b) membrane operators in the checkerboard model. The operators are tensor products of Pauli $X$ or $Z$ over the red qubits. Shaded cubes belong to the $A$ sublattice.

Figure 9

Action of $U$ on the stabilizer generators of $G_{2\mathrm{XC}}$. Here $\prod X(\prod Z)$ denotes the product of Pauli $X$ ($Z$) operators over all depicted qubits. Solid lines correspond to direct lattice edges, whereas dashed lines correspond to dual lattice edges. From top to bottom, the image terms equate to $X_R{X}_G{X}_B{X}_Y$, $X_R{X}_G$, $X_R{X}_B$, $X_R$, $Z_R{Z}_G{Z}_B{Z}_Y$, $Z_R{Z}_G$, $Z_R{Z}_B$, and $Z_R$ operators in the checkerboard model [Eq. (1)], respectively, where $R$, $G$, $B$, and $Y$ refer to the red, green, blue, and yellow cubes.