Fracton Models on General Three-Dimensional Manifolds

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. arXiv:1712.05892v2 [cond-mat.str-el] 30 Dec 2018

Fracton phases of matter

“…(FFO) [34,35]. That is, starting from a model with a larger system size, we can apply a finite depth local unitary transformation and map the model to a smaller system size together with decoupled layers of 2D gapped states, as illustrated in Fig.…”

“…In Ref. 34, this was done for the X-cube model, from which we see that FFO models care not only about the topology of the manifold but also about the foliation structure. On the other hand, it is hard to use the ground state degeneracy as a universal characterization of the order, for reasons explained below.…”

Quantum robustness of fracton phases

“…The X-Cube model, as introduced in [68], is defined on a cubic lattice (for generalization see Refs. [69,70]), where a single spin-1/2 degree of freedom is placed on every edge. The Hamiltonian of the model readŝ…”

“…where L is the circumference of the three-torus. A noncontractible tube of c operators can be written as a product of four such noncontractible lineon loop operators [70] c∈tube c =L…”

Electric circuit realizations of fracton physics

“…This mobility restriction is naturally encoded in the higher moment conservation laws of such systems, such as conservation of dipole moment [23,25,26]. Fractons are notable both for their potential applications to quantum information storage [20,[27][28][29], as well as their prevalence across numerous domains of physics, including spin liquids [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], elasticity [45][46][47][48][49][50][51], localization [19,52,53], hole-doped antiferromagnets [54], gravity [55][56][57], Majorana systems [21,58,59], and deconfined quantum criticality [60].…”