Universal entanglement signatures of foliated fracton phases

Abstract

Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In Ref.[1], we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-partite entanglement quantities, generalizing the proposal of topological entanglement entropy designed for conventional topological phases. We present arguments for the universality of these quantities and show that they attain nonzero constant value in non-trivial foliated fracton phases. arXiv:1803.10426v3 [cond-mat.str-el]

Foliated fracton order in the Majorana checkerboard model

“…As discussed in Ref. 26, the solid torus scheme serves as a diagnostic of the underlying foliation structure, and indeed the result is consistent with the triple foliation structure composed of 2D toric code layers identified in the RG transformation of the section prior. On the other hand, the wireframe scheme is engineered such that the contributions from the foliating layers completely cancel, resulting in a constant value which characterizes the foliated fracton phase.…”

“…Two entanglement schemes, solid torus and wireframe, among others, have proven useful in characterizing the order in foliated fracton models. 26,29,30 In each scheme the quantity to be computed is the conditional mutual information…”

“…The identification of this structure has shed light on the scaling of ground space growth, the structure of fractional excitations in such models, and entanglement entropic signatures discussed previously in the literature. 26,[29][30][31] Furthermore, a more coarse notion of gapped phases of matter is motivated by this observation: in particular, a foliated fracton phase is defined as an equivalence class of Hamiltonians under adiabatic deformation augmented with the possible addition of layers of 2D topological orders.…”

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.…”

“…For example, Ref. 35 proposed an entanglement measure which cancels both the area law and the sub-leading linear part of the entanglement entropy of a sub-region in a gapped 3D model and retains a constant term. The area law term is generic in a gapped system and depends on details of the model, hence non-universal.…”