Searching for fracton orders via symmetry defect condensation

We propose a set of constraints on the ground-state wave functions of fracton phases, which provide a possible generalization of the string-net equations used to characterize topological orders in two spatial dimensions. Our constraint equations arise by exploiting a duality between certain fracton orders and quantum phases with “subsystem” symmetries, which are defined as global symmetries on lower-dimensional manifolds, and then studying the distinct ways in which the defects of a subsystem symmetry group can be consistently condensed to produce a gapped, symmetric state. We numerically solve these constraint equations in certain tractable cases to obtain the following results: in $d=3$ spatial dimensions, the solutions to these equations yield gapped fracton phases that are distinct as conventional quantum phases, along with their dual subsystem symmetry-protected topological (SSPT) states. For an appropriate choice of subsystem symmetry group, we recover known fracton phases such as Haah's code, along with new, symmetry-enriched versions of these phases, such as nonstabilizer fracton models which are distinct from both the X-cube model and the checkerboard model in the presence of global time-reversal symmetry, as well as a variety of fracton phases enriched by spatial symmetries. In $d=2$ dimensions, we find solutions that describe new weak and strong SSPT states, such as ones with both linelike subsystem symmetries and global time-reversal symmetry. In $d=1$ dimension, we show that any group cohomology solution for a symmetry-protected topological state protected by a global symmetry, along with lattice translational symmetry necessarily satisfies our consistency conditions.

Figure 1

Subsystem symmetries act on lower-dimensional regions of the full lattice. Here, a two-dimensional square lattice is shown, with subsystem symmetries acting as spin flips over individual horizontal (blue) or vertical (red) lines. In three spatial dimensions, many quantum phases with subsystem symmetries are known to be dual to models with fracton topological order [1].

Figure 2

Symmetry defects in an SSPT with planar symmetries. We consider an SSPT state protected by planar ${\mathbb{Z}}_2$ symmetries along the (100), (010), and (001) planes along with the translation group of the face-centered cubic (FCC) lattice in (a). The defects of this symmetry group live on the sites of a dual cubic lattice, and a possible configuration of defects is shown in (b). These defects are constrained so that they can only be created in groups of eight at the corners of each of the colored cubes in (b), and quantum fluctuations hop such defects around the lattice. The number of defects in any configuration is always a multiple of four. For the time reversal enriched checkerboard model, which is given by the nonstabilizer Hamiltonian (84), each defect appears with a phase factor $e^{\pi i/4}$.

Figure 3

Consistency conditions. An example of one of the consistent conditions for fluctuating defects in ${\mathbb{Z}}_2$ planar SSPT with symmetries along the (100), (010), and (001) planes in the FCC lattice, considered in Fig. 2. The defect homology equations requires that $e^{i({\varphi }_1+{\varphi }_2)}=e^{i({\varphi }_3+{\varphi }_4)}$. The wavefunction for the dual fracton phase, given by a superposition of allowed defect configurations with these relative amplitudes, is shown schematically.

Figure 4

Examples of consistency conditions coming from $M_{i,i+1}^2=1$ (left) and ${\left(M_{i-1,i}{M}_{i,i+1}\right)}^2=1$ (right). White and black circles denotes the states 0 and 1, respectively.

Figure 5

The translation-enriched X-cube model, which is dual to stacks of 2D cluster states in the horizontal plane, can be probed via a dislocation defect. A lineon cage (blue) going around a dislocation line (red) as shown creates two fracton-dipoles (arrows) parallel to the dislocation line.