Fracton Models on General Three-Dimensional Manifolds

TL;DR

This paper extends the definition of the X-cube fracton model to general three-dimensional manifolds, revealing its topological features and dependence of ground state degeneracy on manifold topology.

Contribution

It introduces a construction of fracton models on arbitrary 3D manifolds using singular foliations, enhancing the understanding of fracton phases and their topological nature.

Findings

X-cube model can be defined on general 3D manifolds

Ground state degeneracy depends on manifold topology

RG transformation relates system size to layered topological states

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.