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Finite-Temperature Quantum Topological Order in Three Dimensions.

Shu-Tong Zhou¹, Meng Cheng², Tibor Rakovszky^3,4,5

¹Nanjing University, Department of Physics, Nanjing 210093, China.

Physical Review Letters

|August 12, 2025

Summary

This summary is machine-generated.

Researchers discovered a novel quantum topological order at finite temperatures using the fermionic toric code. This system exhibits long-range entanglement, driven by an anomalous two-form symmetry, opening new avenues for studying topological phases in realistic dimensions.

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Area of Science:

Condensed Matter Physics
Quantum Information Theory
Topological Phases of Matter

Background:

Quantum topological order typically exists at absolute zero temperature.
Understanding finite-temperature topological phases is crucial for quantum technologies.
The fermionic toric code is a 3D model with emergent fermionic excitations.

Purpose of the Study:

To identify a three-dimensional system exhibiting quantum topological order at finite, nonzero temperatures.
To investigate the role of anomalous symmetries in creating novel topological states.
To explore the possibility of realizing topological phases beyond absolute zero.

Main Methods:

Analysis of the fermionic toric code, a variant of the 3D toric code.
Investigation of emergent fermionic pointlike excitations.
Characterization of anomalous two-form symmetry and its connection to Wilson loops.

Main Results:

Identification of a 3D system with long-range entanglement at small, nonzero temperatures.
Demonstration that anomalous two-form symmetry imbues thermal states with topological order.
The fermionic toric code hosts a topological phase theoretically predicted to exist only at nonzero temperatures.

Conclusions:

The fermionic toric code provides a concrete example of quantum topological order at finite temperatures.
Anomalous two-form symmetries are key to realizing novel topological phases in realistic dimensions.
This work paves the way for studying quantum topological order in physically relevant systems at nonzero temperatures.