Deciphering the nonlocal entanglement entropy of fracton topological orders

TL;DR

The paper addresses how to quantify nonlocal entanglement in fracton topological orders by introducing $S_{nonlocal}$ as a conditional mutual information based diagnostic. It develops a powerful lower bound under Abelian stabilizer assumptions by exploiting condensate and deformable operator structures and shows how the bound saturates for conventional topological orders while becoming geometry dependent for fracton phases. Applying the bound to the 2D and 3D toric codes, the X-Cube model, and fractal spin liquids, the authors demonstrate saturation in conventional orders and extensive, geometry-sensitive $S_{nonlocal}$ in fracton models. The results provide a practical tool to distinguish fracton orders, illuminate the role of nonlocal condensates, and discuss robustness under local perturbations, with implications for broader understanding of entanglement in exotic quantum phases.

Abstract

The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy $S_{topo}$ for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy $S_{nonlocal}$ (a generalization of $S_{topo}$). The lower bound applies to Abelian stabilizer models including conventional topological orders as well as type \Rom{1} and type \Rom{2} fracton models, and it could be used to distinguish them. For fracton models, the lower bound shows that $S_{nonlocal}$ could obtain geometry-dependent values, and $S_{nonlocal}$ is extensive for certain choices of subsystems, including some choices which always give zero for TQFT. The stability of the lower bound under local perturbations is discussed.

Figures (14)

• Figure 1: A system is divided into subsystems $A,B,C$ and $D$. Geometrically and topologically distinct choices will be used throughout the paper. They share the following features: $\partial A\cap\partial C=0$ and other pairs of subsystems have shared boundaries.
• Figure 2: Condensate operators from different topological orders $W$, $W'$, $W"$ (in orange color), and the truncations of corresponding operators give us deformable operators $U$, $U'$, $U"$ (in blue color) which create topological excitations (in red color) when acting on the ground state. The operators $W$, $U$ are from the 2D toric code model; the operators $W'$, $U'$ are from the 3D toric code model; the operators $W"$, $U"$ are from a fractal spin liquid model (note that $W"$ is not a precise depiction). The support of $W$ is a closed loop; the support of $W'$ is a closed membrane, i.e. the 2D boundary of the box; the support of $W"$ has fractal structure. The picture only shows part of $W"$ explicitly, the rest of $W"$ is embedded in the dotted 2D surfaces with dashed 1D edges, and it may contain fractal parts or 1D parts.
• Figure 3: An illustration of the support of different operators: $W_i$ (in orange color) is supported on $ABC$; $U_i$ (in blue color) is supported on $CD$; $U_i^{def}$ (in blue color) is supported on $AD$. The topological excitations created by $U_i$ or $U_i^{def}$ are shown in red color. The color setting of the operators and excitations will be used throughout the paper. We do not assume these operators to be integer dimensional and the construction applies to models in different dimensions.
• Figure 4: Deformable operators $U_1$, $U_2$ supported on open strings and condensate operators $W_1$, $W_2$ supported on closed strings. $U_1$, $U_2$ create topological excitations around their endpoints. In other words, $U_1$ flips two plaquettes and $U_2$ flips two stars.
• Figure 5: Topological excitations in the 3D toric code model and their detection using condensates.