Fracton Models on General Three-Dimensional Manifolds

TL;DR

This work extends the X-cube fracton model from cubic lattices to general closed 3-manifolds by introducing singular compact total foliations (SCTFs) that partition a manifold into intersecting leaf stacks. A renormalization perspective shows that ground-state degeneracy is governed by leaf topology via a relation $\log_2 \mathrm{GSD} = b_x L_x + b_y L_y + b_z L_z - c$, interpretable as adding or removing 2D toric-code layers. The authors demonstrate explicit manifold constructions (including $S^2\times S^1$, $S^3$, half-twist, Klein bottle times $S^1$, and $\Sigma_g\times S^1$) with corresponding GSDs, clarify how foliations can induce or remove localized logical operators, and connect the fracton phase to an entanglement RG framework, ultimately proposing a SCTF-TQFT viewpoint for type-I fracton orders. They extend the construction to $ ext{Z}_N$ generalizations and relate the RG to the Haah code, offering a unified picture in which 2D topological layers serve as computationally trivial resources within a broader 3D fracton phase landscape. Overall, the paper broadens the topology-dependent understanding of fracton orders and provides a concrete framework for their continuum-like description and generalizations to other manifolds and group structures.

Abstract

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 only been defined and analyzed 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.

Figures (11)

• 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 $W^z_{mn}$. 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 endpoints (black dots).
• Figure 4: A construction with periodically placed spheres. (Sec. \ref{['sec:spheres']}). (a-b) We place spheres of radius 0.46 on an face-centered cubic (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 3-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 8 spherical leaves of constant $x$, $y$, 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 $180\degree$ 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 3-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. Hexahedral. A $\Sigma_2$ cross-section of a cellulation of $\Sigma_2\times S^1$. The red and blue lines correspond to leaves of respective singular foliations. The singularities are indicated by the black lines.