Detecting Topological Order at Finite Temperature Using Entanglement Negativity

We propose a diagnostic for finite temperature topological order using "topological entanglement negativity," the long-range component of a mixed-state entanglement measure. As a demonstration, we study the toric code model in d spatial dimensions for d=2,3,4, and find that when topological order survives thermal fluctuations, it possesses a nonzero topological entanglement negativity, whose value is equal to the topological entanglement entropy at zero temperature. Furthermore, we show that the Gibbs state of 2D and 3D toric code at any nonzero temperature, and that of 4D toric code above a certain critical temperature, can be expressed as a convex combination of short-range entangled pure states, consistent with the absence of topological order.