Universal entanglement signatures of foliated fracton phases

TL;DR

The paper addresses the challenge of identifying universal characteristics of foliated fracton phases, a generalization of gapped phases that allows 2D topological layers in adiabatic transformations between 3D gapped Hamiltonians. It develops universal, entanglement-based signatures by extending topological entanglement entropy concepts to foliated fracton order and introducing wireframe entanglement schemes that cancel area-law and foliation-layer contributions, revealing nonzero constants in nontrivial phases such as the X-cube and related models. The authors derive criteria for universality, compute lower bounds on conditional mutual information, and show how orientation and foliation structure influence the measured quantities, thereby providing a diagnostic toolkit for foliated fracton order. This work advances the understanding of long-range entanglement in fracton systems and lays groundwork for connecting foliated fracton phases to higher-rank gauge theories, with open questions about a complete universal classification and extensions to type-II and gapless fracton models.

Abstract

Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In a previous work, 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 non-zero constant value in non-trivial foliated fracton phases.

Figures (7)

• Figure 1: (a) Square $I(A;B;C)$ and (b) annular $I(A;B|C)$ schemes to isolate topological entanglement entropy in 2D.
• Figure 2: (a) 3D cube $I(A;B;C)$ and (b) solid torus $I(A;B|C)$ schemes. In both cases the regions are contained within an overall cube of side length $L$.
• Figure 3: (a) Cubic $I(A;B|C)$, (b) cubic $I(A;B;C;D|E)$, (c) triangular prism $I(A;B|C)$, and (d) tetrahedral $I(A;B|C)$ entanglement schemes for foliated fracton phases. (d) Stabilizer for the Chamon model defined on a cubic lattice with one qubit per vertex (inset).
• Figure 4: A side-view of the cubic entanglement regions (green) from Fig. \ref{['fig:wireframe']} for different possible orientations with respect to the foliating layers (red). (a) Proper alignment on a cubic lattice, yielding the values Table \ref{['tab:entropy']}. (b) Improper alignment, for which entanglement quantities $I(A;B|C) = I(A;B;C;D|E)=0$. (c) Top-down view of a properly aligned solid wireframe on a stacked-kagome lattice, which yields $I(A;B|C) = I(A;B;C;D|E)=1$ for the kagome X-cube model Slagle17Lattices as per Table \ref{['tab:entropy']}.
• Figure 5: Operators satisfying the conditions of Eq. (\ref{['eq:bound conditions']}) which can be used to bound $I(A;B|C)$ for the cubic scheme depicted in Fig. \ref{['fig:wireframe']}. (a) For the X-cube model (Fig. \ref{['fig:models']}), $I(A;B|C) \geq 1$ is obtained by taking $W_1$ to be a product of $X$ operators along the blue lines, and $U_1$ and $U_1^\text{def}$ to be products of $Z$ operators over all links that penetrate the red and yellow regions, respectively. For the checkerboard model (Fig. \ref{['fig:models']}), $I(A;B|C) \geq 2$ can be obtained by taking $W_1$ ($W_2$) to be a product of $X$ ($Z$) operators along the blue lines, and $U_1$ and $U_1^\text{def}$ ($U_2$ and $U_2^\text{def}$) to be products of $Z$ ($X$) operators over the red and yellow surfaces, respectively. $I(A;B|C) \geq 1$ can similarly be obtained for the Chamon model (Fig. \ref{['fig:models']}) using a tetrahedal-shaped geometry, but each operator will contain a mix of $X$, $Y$, and $Z$ Pauli operators. (b) Another view with subsystem $C$ (green) hidden for clarity.