Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface

Abstract: Topologically-ordered phases of matter at non-zero temperature are conjectured to exhibit universal patterns of long-range entanglement which may be detected by a mixed-state entanglement measure known as entanglement negativity. We show that the entanglement negativity in certain topological orders can be understood through the properties of an emergent symmetry-protected topological (SPT) order which is localized on the entanglement bipartition. This connection leads to an understanding of ($i$) universal contributions to the entanglement negativity which diagnose finite-temperature topological order, and ($ii$) the behavior of the entanglement negativity across certain phase transitions in which thermal fluctuations eventually destroy long-range entanglement across the bipartition surface. Within this correspondence, the universal patterns of entanglement in the finite-temperature topological order are related to the stability of an emergent SPT order against a symmetry-breaking field. SPT orders protected by higher-form symmetries -- which arise, for example, in the description of the entanglement negativity for $\mathbb{Z}_{2}$ topological order in $d=4$ spatial dimensions -- remain robust even in the presence of a weak symmetry-breaking perturbation, leading to long-range entanglement at non-zero temperature for certain topological orders.