Topologically ordered phases are exotic low-temperature phases of matter. They are of interest both from a fundamental scientific perspective, because they exhibit unusual properties such as excitations that are neither bosons nor fermions, and from an applied perspective, because they may be useful for quantum error correction and topological quantum computing. The simplest model with topological order is the toric code, which plays a central role in quantum error correction.
In a recent paper, Kai-Hsin Wu et al. enriched the toric code with an additional U(1) symmetry and provided numerical evidence that the system is topologically ordered. By further generalizing this model to the U(1) checkerboard toric code, Maximilian Vieweg, Kai Phillip Schmidt, and others use perturbation theory to provide compelling evidence that the phase is not topologically ordered, but nevertheless has low-energy excitations with fascinating properties. Their mobility is restricted: an excitation in isolation cannot move without creating other excitations. Such excitations are called fractons. In the case of the U(1) checkerboard toric code, these fractons are confined and cannot exist in isolation.
For more information, see their publication in SciPost Physics:
Absence of topological order in the U(1) checkerboard toric code
Maximilian Vieweg, Viktor Kott, Lea Lenke, Andreas Schellenberger, Kai Phillip Schmidt
SciPost Phys. 20, 056 (2026)
Topologically ordered phases are exotic low-temperature phases of matter. They are of interest both from a fundamental scientific perspective, because they exhibit unusual properties such as excitations that are neither bosons nor fermions, and from an applied perspective, because they may be useful for quantum error correction and topological quantum computing. The simplest model with topological order is the toric code, which plays a central role in quantum error correction.
In a recent paper, Kai-Hsin Wu et al. enriched the toric code with an additional U(1) symmetry and provided numerical evidence that the system is topologically ordered. By further generalizing this model to the U(1) checkerboard toric code, Maximilian Vieweg, Kai Phillip Schmidt, and others use perturbation theory to provide compelling evidence that the phase is not topologically ordered, but nevertheless has low-energy excitations with fascinating properties. Their mobility is restricted: an excitation in isolation cannot move without creating other excitations. Such excitations are called fractons. In the case of the U(1) checkerboard toric code, these fractons are confined and cannot exist in isolation.
For more information, see their publication in SciPost Physics:
Absence of topological order in the U(1) checkerboard toric code
Maximilian Vieweg, Viktor Kott, Lea Lenke, Andreas Schellenberger, Kai Phillip Schmidt
SciPost Phys. 20, 056 (2026)