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

Published on: June 7, 2018

Topological order at nonzero temperature.

Matthew B Hastings1

  • 1Duke University, Department of Physics, Durham, North Carolina, 27708, USA.

Physical Review Letters
|December 21, 2011
PubMed
Summary
This summary is machine-generated.

We define topological order at nonzero temperatures, showing that simple 2D systems are not topologically ordered. This finding impacts quantum information storage, though complex systems like the toric code remain nontrivial.

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Last Updated: May 26, 2026

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Area of Science:

  • Condensed Matter Physics
  • Quantum Information Theory

Background:

  • Topological order is typically defined at zero temperature.
  • Understanding topological order at nonzero temperatures is crucial for robust quantum technologies.

Purpose of the Study:

  • To propose a definition for topological order at nonzero temperatures.
  • To investigate the topological properties of quantum states in the presence of thermal fluctuations.

Main Methods:

  • Defining topological order via local quantum circuits.
  • Analyzing Hamiltonians composed of commuting local terms in 2D.
  • Examining quantum information storage using stringlike operators.

Main Results:

  • Any 2D Hamiltonian that is a sum of commuting local terms is not topologically ordered at T > 0.
  • These trivial states cannot store quantum information via specific stringlike operators.
  • The four-dimensional toric code exhibits a nontrivial phase at nonzero temperatures.

Conclusions:

  • The proposed definition allows for nontrivial topological phases at nonzero temperatures.
  • The findings have implications for the stability and application of topological quantum states in realistic (non-zero temperature) environments.