Detecting Topological Order at Finite Temperature Using Entanglement Negativity

We propose a scheme to diagnose finite temperature topological order using the long-range component of entanglement negativity, dubbed topological entanglement negativity. As a demonstration, we study the toric code model in d spatial dimension for d=2,3,4, and find that a topological ordered state at finite temperature has non-zero topological entanglement negativity, whose value is equal to topological entanglement entropy at zero temperature. To calculate entanglement negativity, we develop a general tool for any commuting projector Hamiltonians to derive the spectrum of a thermal state under partial transpose, allowing us to map the calculation of negativity to a classical statistical mechanics problem. Relatedly, using the idea of minimally entangled typical thermal states, we derive necessary conditions for the existence of finite temperature topological order in any CSS code Hamiltonians.