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.