Image depicting the phase transition that occurs in our model as a function of the temperature. The model has both point-like excitations and loop-like excitations. At low temperatures, the point-like excitations proliferate — the typical thermal fluctuations explore many possible configurations of the point-like excitations. For the loop-like excitations on the other hand, at low temperatures, they are confined — only small loop configurations are typical. The 3D phase of matter that we identified lives at low temperatures and has small loop configurations, typically. In contrast, at high temperatures, the point-like excitations and the loop-like excitations proliferate. It is now typical to find configurations with large loop excitations. Credit: Zhou et al.
Some phases of matter cannot be described using the conventional framework of symmetry breaking and exhibit a so-called quantum order. One type of quantum order, known as topological order, is characterized by long-range entanglement between particles across an entire system, a ground state degeneracy that depends on the global shape of the system, and a robustness against local disturbances.
Topological phases of matter primarily occur at zero temperature, as thermal fluctuations tend to destroy them and disrupt their underlying order. In a recent paper in Â鶹ÒùÔºical Review Letters, however, researchers at Nanjing University, Yale University and other institutes reported a new 3D topological phase of matter characterized by an anomalous two-form symmetry that occurs at non-zero temperatures.
"In the last several years, we have made substantial progress in our ability to control quantum systems—over a range of different platforms: superconducting qubits, trapped ions, neutral atoms, photonics, and so on," Tyler D. Ellison, senior author of the paper, told Â鶹ÒùÔº.
"This has opened up the potential for engineering interesting quantum states and quantum phases of matter in well-controlled experimental settings. However, unavoidably, the hardware is imperfect, and the system is not completely isolated from the environment. This means that the quantum system suffers from faulty operations (such as photon loss) and noise from the environment (e.g., from cosmic-ray particles)."
Due to this key limitation, quantum systems are not defined by 'pure' quantum states occurring in them, but rather by probability distributions of quantum states, arising from the probabilistic nature of noise-associated errors. Over the past decades, many physicists have been trying to better understand exotic physical phenomena and quantum states of matter that can be realized irrespective of background noise and a lack of control over them.
"Similarly, quantum systems at non-zero temperature are probability distributions of pure quantum states," said Ellison. "In this case, the distributions instead arise from thermal fluctuations. We realized that some of the same theoretical tools that have been developed recently in the context of noisy quantum systems can be used to characterize quantum systems at non-zero temperature."
A graphical representation of the anomalous symmetry. Unlike a translation or rotation symmetry, the symmetry acts only on degrees of freedom along loops. Intuitively, it detects the number of loop-like excitations that it links with. At low temperatures, the loops are small, and the quantum state is approximately symmetric. Credit: Zhou et al.
When they first started conducting their recent study, Ellison and his colleagues had not set out to uncover new 3D quantum phases of matter, but were instead exploring quantum phases of matter arising in "noisy" systems. Nonetheless, their efforts ultimately led to the discovery of a new topological order at non-zero temperatures, dubbed the fermionic toric code.
"It has long been appreciated that symmetries play an important role in characterizing phases of matter," explained Ellison. "Liquids have a continuous translational symmetry, since there is a uniform probability of finding an atom anywhere in the system, while solids have a discrete translational symmetry, due to the crystal structure formed by the atoms.
"More exotic phases of matter are likewise distinguished by their more exotic symmetries. Some symmetries, known as anomalous symmetries, are exotic enough that, if any system possesses the symmetry, then it must be highly entangled."
As part of their study, Ellison and his colleagues identified a model for a 3D system that exhibits an anomalous symmetry at non-zero temperatures. The model in question, known as the fermionic toric code, is a variant of the well-known toric code, a model typically employed to perform quantum error correction and topological quantum computations.
They showed that this model exhibits an anomalous two-form symmetry at temperatures above absolute zero. Based on their analyses, the researchers argued that this system must be highly entangled at low temperatures.
"It came as a complete surprise to us that there exists a quantum phase of matter in three dimensions at non-zero temperature," said Ellison.
"There are strong arguments to say that no such quantum phases of matter exist in 2D, and prior to our work, the general expectation in the community was that the arguments hold in 3D. On the other hand, it has been known that there are quantum phases of matter in four spatial dimensions that exist at non-zero temperature. However, these are a bit of a fantasy, since the interactions engineered in experimental settings are in three spatial dimensions."
This recent study introduces the first-ever example of a quantum phase of matter that could realistically be leveraged to design quantum systems at equilibrium. As part of their future studies, Ellison and his colleagues plan to further investigate the newly uncovered phase of matter they uncovered and explore its possible practical applications.
"Our next step will be to study how to engineer our model on an experimental platform. We expect that arrays of neutral atoms are a natural fit for the interactions in our model.
"Another important direction is for us to develop simple diagnostics to detect whether we have successfully prepared the phase of matter. Once we have an experimental realization, it opens the door to exploring the exotic properties of quantum phases of matter at non-zero temperatures," added Ellison.
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More information: Zhou et al, Finite-Temperature Quantum Topological Order in Three Dimensions, Â鶹ÒùÔºical Review Letters (2025). . On arXiv:
Journal information: Â鶹ÒùÔºical Review Letters , arXiv
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