Foliated fracton order in the checkerboard model

TL;DR

The paper establishes that the checkerboard fracton model resides in a foliated fracton phase and is equivalent to two copies of the X-cube model, using a toric-code bilayer renormalization group to realize a fixed-point under foliated equivalence. By analyzing ground-state entanglement, quotient superselection sectors, and an explicit local unitary mapping (with ancillas) to two X-cube copies, it demonstrates that universal foliated fracton data match those of the two-XCube system. The generalization to arbitrary 3-manifolds via SCTF and the entanglement-entropy schemes provide robust, foliations-free diagnostics for fracton order, decoupled from 2D layer resources. Overall, the work clarifies how checkerboard order fits within foliated fracton classification and offers a concrete methodology for comparing foliated fracton phases across models and geometries.

Abstract

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.

Figures (9)

• Figure 1: (a) $A$-$B$ 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 $\ket{\psi_\mathrm{CB}}\otimes\ket{\psi_{\mathrm{TC}}^a}\otimes\ket{\psi_{\mathrm{TC}}^b}$. Here $\prod X$$\left(\prod Z\right)$ 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.