Detecting Topological Order at Finite Temperature Using Entanglement Negativity

TL;DR

This paper introduces a new method using topological entanglement negativity to detect finite temperature topological order, demonstrating its effectiveness on the toric code model across different dimensions.

Contribution

It proposes a diagnostic tool based on entanglement negativity for finite temperature topological order and applies it to the toric code in various dimensions.

Findings

Non-zero topological entanglement negativity indicates surviving topological order.

Gibbs states at non-zero temperature can be decomposed into short-range entangled states.

Topological order disappears above a critical temperature in 4D toric code.

Abstract

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 dimension for $d$=2,3,4, and find that when topological order survives thermal fluctuations, it possesses a non-zero 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 non-zero 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.