Compactifying fracton stabilizer models

We investigate two-dimensional compactifications of three-dimensional fractonic stabilizer models. We find the two-dimensional topological phases produced as a function of compactification radius for the $X$-cube model and Haah's cubic code. Furthermore, we uncover translation symmetry enrichment in the compactified cubic code that leads to twisted boundary conditions.

Figure 1

Structure theorem [90, 91, 92]: a translation invariant topological stabilizer Hamiltonian in 2D is equivalent to copies of toric code and a trivial state via a locality-preserving unitary. This can be used to decompose a compactified 3D model (left) into copies of the 2D toric code (right).

Figure 2

A section of a 2D stabilizer model is shown, with circles depicting the qubits. Two strips are shown, with solid blue bars on the ends denoting open boundary conditions and thin lines denoting closed boundary conditions. Half the rank of the commutation matrix between operators on the strips gives the number of copies of toric code.

Figure 3

Regions used to calculate the topological entanglement entropy.

Figure 4

Excitation patterns of stabilizers located on the vertices of the dual cubic lattice, which are created by local operators on pairs of qubits inside each cube: (a) $XI$, (b) $IX$, (c) $XX$, (d) $IX$–$XI$ along $\widehat{z}$, and (e) $IX$–$XX$–$XI$ along $\widehat{z}$.