Systematic construction of gapped nonliquid states

Using Abelian and non-Abelian topological orders in two-dimensional (2D) space and the different ways to glue them together via their gapped boundaries, we propose a systematic way to construct three-dimensional (3D) gapped states (and in other dimensions). The resulting states are called cellular topological states, which include gapped nonliquid states, as well as gapped liquid states in some special cases. Some new fracton states with fractal excitations are constructed even using 2D $Z_2$ topological order. More general cellular topological states can be constructed by connecting 2D domain walls between different 3D topological orders. The constructed cellular topological states can be viewed as fixed-point states for a reverse renormalization of gapped nonliquid states.

Figure 1

The 3D space is divided into cells. Each cell surface is occupied by a $2+1\mathrm{D}$ topological order. Each edge is occupied by an anomalous $1+1\mathrm{D}$ topological order, which is a gapped boundary of a stacking of the $2+1\mathrm{D}$ topological orders.

Figure 2

The 3D space is decomposed into the hexagonal column's. The honeycomb lattice has two kinds of vertices: Type-A vertices have arrows pointing in, and type-B vertices have arrows pointing out.

Figure 3

For ${\mathcal{C}}_i$ satisfying Eq. (2), the layer ${\mathcal{M}}_{ij}$ is detached.

Figure 4

(a) A deformation step $\mathcal{C}{⊠}_{\mathcal{M}}\mathcal{D}={\mathcal{C}}^{\prime }{⊠}_{\tilde{\mathcal{M}}}{\mathcal{D}}^{\prime }$ [see Eq. (11)]. (b) Using the deformation step, we can change the blue-hexagonal tensor network to the one formed by red links and light-blue dots. (c) Shrinking the triangles to the red dots produces the blue-hexagonal tensor network [see Eq. (12)]. This completes a renormalization step $(\mathcal{M},\mathcal{C},\mathcal{D})\to (\tilde{\mathcal{M}},\tilde{\mathcal{C}},\tilde{\mathcal{D}})$.

Figure 5

(a) Fusion of two $1+1\mathrm{D}$ domain walls $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$ connected by a $2+1\mathrm{D}$ topological order ${\mathcal{M}}^{\prime }$ gives rise to ${\mathcal{C}}^{\prime \prime }={\mathcal{C}}^{\prime }{⊠}_{{\mathcal{M}}^{\prime }}\mathcal{C}$. The looplike $t$ direction is not shown. (b) Exchanging $x$ and $t$, we get the corresponding wave function overlaps. Two wave-function overlaps $C$ and $C^{\prime }$ can be reduced to one wave-function overlap $C^{\prime \prime }$. The looplike $x$ direction is not shown.

Figure 6

A simple tensor network formed by two vertices connected by a link. The link corresponds to a $2+1\mathrm{D}$ topological order $\mathcal{M}$. The vertices corresponds to a $1+1\mathrm{D}$ anomalous topological order $\mathcal{C},\mathcal{D}$.

Figure 7

If we choose $\tilde{\mathcal{M}}=\mathcal{M}⊠\mathcal{M}$, then Eq. (11) always has solutions.

Figure 8

In the cellular topological state $({\mathcal{G}}_{Z_2}^{2+1},{\mathcal{C}}_1,{\mathcal{C}}_1)$, the $e$ particle can move freely in 3D space. The unmarked links are in sector 1. A configuration with a loop of links in sector 2 (marked by $e-e$) corresponds to another degenerate ground state.

Figure 9

In the cellular topological state $({\mathcal{G}}_{Z_2}^{2+1},{\mathcal{C}}_1,{\mathcal{C}}_1)$, the $m$ particles must form a closed loop in the dual honeycomb lattice.

Figure 10

In the cellular topological state $({\mathcal{G}}_{Z_2}^{2+1},{\mathcal{C}}_2,{\mathcal{C}}_2)$, the $f$ particle can move freely in 3D space.

Figure 11

In the cellular topological state $({\mathcal{G}}_{Z_2}^{2+1},{\mathcal{C}}_1,{\mathcal{C}}_3)$, the $e$ particle can move across the ${\mathcal{C}}_1$ boundary, and the $m$ particle can move across the ${\mathcal{C}}_3$ boundary. However, the long distant motion of $e$ and $m$ particles are blocked.

Figure 12

A configuration in the ground state. The unmarked links are in sector 1. The links marked by $m-m$ are in sector 3.

Figure 13

A configuration of the cellular topological state. The unmarked links are in sector 1. The links marked by $m-m$ are in sector 3. Only the vertices in the blue circle cost energy.

Figure 14

A cellular topological state on cubic lattice. The square faces are occupied by $2+1\mathrm{D}$ $Z_2$ topological order, and the red, blue, and green lines correspond to three kinds of $1+1\mathrm{D}$ anomalous topological orders ${\mathcal{C}}_R$, ${\mathcal{C}}_B$, and ${\mathcal{C}}_G$.