Understanding entanglement negativity in topological order at finite temperature

It has been proposed that the stability of topologically ordered states of matter at finite temperature can be diagnosed by their long-range entanglement structure using entanglement negativity, a mixed-state entanglement measure. In this work, we provide a novel connection between entanglement negativity in a topological order and an emergent symmetry-protected topological (SPT) order localized on the entanglement bipartition. This connection leads to a precise understanding of the phase transition in entanglement negativity as the temperature is increased, and as thermal fluctuations eventually destroy the long-range entanglement in the topological phase. Within this correspondence, anyons in topological order correspond to symmetry charges in SPT order, and the stability of topological order at a non-zero temperature relates to the stability of SPT order against a symmetry-breaking field. For the 4d toric code and 3d toric code with point-like charges forbidden, in which topological order exists at finite temperature, the corresponding SPT order is protected by a higher-form symmetry that is robust under a weak symmetry-breaking field. Finally, a universal scaling form of long-range entanglement negativity is derived across the finite temperature transition of topological order.