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Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses

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Detecting Topological Order at Finite Temperature Using Entanglement Negativity.

Tsung-Cheng Lu¹, Timothy H Hsieh², Tarun Grover¹

¹Department of Physics, University of California at San Diego, La Jolla, California 92093, USA.

Physical Review Letters

|September 25, 2020

Summary

We introduce topological entanglement negativity to diagnose topological order at finite temperatures. This new measure detects topological order in the toric code model when it persists despite thermal fluctuations.

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Area of Science:

Condensed Matter Physics
Quantum Information Theory
Topological Quantum Matter

Background:

Topological order describes exotic quantum phases of matter characterized by long-range entanglement.
Distinguishing topological order from thermal states at finite temperatures remains a challenge.
Entanglement measures are crucial for characterizing quantum phases.

Purpose of the Study:

To propose and demonstrate a new diagnostic tool for identifying topological order in mixed quantum states at finite temperatures.
To investigate the behavior of topological entanglement negativity in the toric code model across different spatial dimensions.

Main Methods:

Utilizing topological entanglement negativity, a component of mixed-state entanglement measures.
Analyzing the toric code model in d=2, 3, and 4 spatial dimensions.
Examining the Gibbs state of the toric code at various temperatures.

Main Results:

Topological entanglement negativity is non-zero when topological order survives thermal fluctuations in the toric code.
The value of topological entanglement negativity equals the zero-temperature topological entanglement entropy.
Gibbs states of 2D and 3D toric code (and 4D above a critical temperature) are convex combinations of short-range entangled states, indicating absence of topological order.

Conclusions:

Topological entanglement negativity serves as a robust diagnostic for finite-temperature topological order.
The study provides insights into the thermal stability and characterization of topological phases.
Results confirm the expected loss of topological order in the toric code model under specific thermal conditions.