Fracton Models on General Three-Dimensional Manifolds

Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some “topological” features: They support fractional bulk excitations (dubbed fractons) and a ground-state degeneracy that is robust to local perturbations. However, because previous fracton models have been defined and analyzed only on a cubic lattice with periodic boundary conditions, it is unclear to what extent a notion of topology is applicable. In this paper, we demonstrate that the $X$-cube model, a prototypical type-I fracton model, can be defined on general three-dimensional manifolds. Our construction revolves around the notion of a singular compact total foliation of the spatial manifold, which constructs a lattice from intersecting stacks of parallel surfaces called leaves. We find that the ground-state degeneracy depends on the topology of the leaves and the pattern of leaf intersections. We further show that such a dependence can be understood from a renormalization group transformation for the $X$-cube model, wherein the system size can be changed by adding or removing 2D layers of topological states. Our results lead to an improved definition of fracton phase and bring to the fore the topological nature of fracton orders.

Popular Summary

Topological phases of matter have an amazing sensitivity to the global shape of the space in which the system is embedded, i.e., the topology of the spatial manifold. Recently, researchers discovered a new class of models beyond topological phases dubbed fracton models, which share many similarities with topological phases but appear to be sensitive to the underlying manifold in fundamentally different ways. This challenged preconceptions of what types of quantum phases are physically possible. Here, we develop a more systematic theoretical understanding of fracton models, which could help researchers understand the exotic properties of these models, such as their capacity to store quantum information.

We show that the important feature to which fracton models are sensitive is the “foliation structure,” a concept used in geometry and geology to describe the slicing up of a manifold with parallel surfaces. For example, horizontal planes stacked vertically provide one way to foliate 3D space. Fracton models can respond to changes in the spatial manifold’s foliation structure, which underlies the structure of the crystal lattice, by changing the ground-state degeneracy.

The connection to foliation has led us to concretely define a “foliated fracton phase” as a 3D quantum phase where the system size may be increased by “sewing” in additional layers of 2D topological phases. This perspective not only helps clarify the relation between various toy models discovered so far but also expands the discussion into more realistic and maybe even experimentally accessible models.

Figure 1

(a) Cube and (b) cross operators of the $X$-cube model Hamiltonian on a cubic lattice.

Figure 2

Visualization of logical operators. The green string corresponds to $W_{mn}^z$. The product of the four operators corresponding to the blue strings is equal to the identity, as described in the main text.

Figure 3

Visualization of particle creation operators. The red links correspond to a membrane geometry on the dual lattice. The product of $Z$ operators over these edges excites the (darkened) cube operators at the corners. The product of $X$ operators over the links comprising the straight open blue string creates excitations at its end points (black dots).

Figure 4

A construction with periodically placed spheres (see Sec. 3b). (a),(b) We place spheres of radius 0.46 on fcc lattice. The spheres in (b) are located at the blue points of the fcc lattice in (a). When the $X$-cube model is defined on the resulting lattice, the phase is equivalent to the 3D toric code. (b) The toric code charges reside on small cubes. These charges can hop, e.g., between the two blue cubes via a string of $Z$ operators on the two red edges. (c) The elementary three-cells of the cellulation. (d),(e) Membrane and string operators. The membrane operator is a product of $X$ operators on the blue edges, whereas the string operator is a product of $Z$ operators on the red edges.

Figure 5

(a) A spherical cross section of a cellulation of $S^2\times S^1$ with $L_x=L_y=8$. (b) The $t=0$ equator of $S^3$ defined as the locus of points in ${\mathbb{R}}^4$ satisfying $x^2+y^2+z^2+t^2=1$. In this example, $S^3$ is foliated by eight spherical leaves of constants $x$, $y$, and $z$, which are colored red, green, and blue. Although the sphere drawn in (a) is a leaf, the sphere drawn in (b) is not a leaf; it is merely a convenient cross section. (c) The half-twist manifold constructed by identifying opposite faces of a cube. The front and back faces are glued after a 180° twist. The dashed red and green squares are outlines of embedded Klein bottles. The pair of solid red (or green) squares outline a single torus, as does the blue square. (d) The three-manifold $K^2\times S^1$ viewed as a cube with opposite faces identified; front and back faces are identified after a reflection across the vertical bisector. The pair of solid red squares outlines a single embedded torus, as do the dashed red square and solid blue square. The solid green square outlines an embedded Klein bottle. (e) Figure courtesy of Ref. [27]. A ${\mathrm{\Sigma }}_2$ cross section of a cellulation of ${\mathrm{\Sigma }}_2\times S^1$. The red and blue lines correspond to the leaves of the respective singular foliations. The singularities are indicated by the black lines.

Figure 6

(a) Adding an $xy$ layer to the $X$-cube model on $T^3$. The large cube represents a unit cell of the original $X$-cube model, while the bold (blue) square is an elementary plaquette of the new layer $\alpha$. The original $z$-oriented edges are split into two by the new layer. The local unitary $S$ is a translation-invariant composition of commuting cnot gates; a unit cell is pictured here. Arrows point from the control qubit to the target qubit. (b) Action of the unitary $S$ on the qubits of a hexagonal prism three-cell. The lower hexagonal plaquette belongs to the new $z$ layer $\alpha$. Bold (blue) edges are transverse to the $y$ foliation, whereas double (green) edges are transverse to the $x$ foliation.

Figure 7

Adjoint action of $S$ on stabilizers of $|{\psi }_{\mathrm{XC}}{⟩}^{\prime }\otimes |{\psi }_{\mathrm{TC}}⟩$. As in Fig. 6, bold (blue) lines correspond to edges of the new layer. Terms not pictured are unchanged.

Figure 8

(a) Action of three-cell operator $B_c$ in the ${\mathbb{Z}}_N$ $X$-cube Hamiltonian on a cubic three-cell. Vertices of the cube are given an $A\text{−}B$ bipartition. (b) Cross-shaped operators $A_v^{\mu }$ of the ${\mathbb{Z}}_N$ $X$-cube model Hamiltonian.

Figure 9

Adding a layer to the ${\mathbb{Z}}_N$ $X$-cube model on a torus, as in Fig. 6. For the ${\mathbb{Z}}_N$ case, $S$ is a translation-invariant product of commuting $C$ and $C^{†}$ operators shown here is a unit cell. Arrows point from control qudit to target qudit; a single shaft indicates $C$, whereas a double shaft corresponds to $C^{†}$.

Figure 10

Stereographically projected spherical cross section of the local unitary operator $S$, which sews the dark blue toric code layer into the $X$-cube lattice, as used in the RG transformation for $S^2\times S^1$. $S$ is a product of cnot gates corresponding to the arrows, which point from a control qubit to a target qubit. The arrows on the edges indicate gates that act on the edges oriented into (and out of) the plane and located at the adjacent vertices. Most of the cnot gates are acting within cubes (depicted as curved squares above); within these cubes, $S$ is the same as in Fig. 6. The toric code plaquette operators extending out of the plane from the dashed blue lines are mapped to composite three-cell operators on the three-cells extending out of the plane from the shaded light blue region.

Figure 11

(a) The equatorial cross section of $S^3$ from Fig. 5. We emphasize that the sphere drawn in (a) is not a leaf; it is merely a convenient cross section. (b),(c) Stereographic projections of spherical leaves embedded in $S^3$. (b) and (c) intersect the equator [i.e., the spherical cross section shown in (a)] at the solid and dashed red lines in (a), respectively. The green and blue lines represent links of the cellulation lying along the respective red colored leaves in (a). The numbered vertices correspond to links that connect the two leaves and, thus, share an index in (b) and (c). The unitary $S$ sews a toric code state (on the dashed red leaf) into the $X$-cube model on $S^3$. $S$ consists of a product of cnot gates and maps toric code plaquette operators on the dashed red layer to composite three-cell operators lying between the two leaves. Some of the faces of these composite operators are shaded in the figures as example. Plaquettes in (b) and (c) with corresponding colors indicate faces that belong to the same composite three-cell.