- Accepted Paper
Hierarchy of mixed symmetry protected topological states in extended cluster states under subsystem decoherence
Phys. Rev. B - Accepted 28 October, 2026
DOI: https://doi.org/10.1103/3tn7-pqj4
Phys. Rev. B - Accepted 28 October, 2026
DOI: https://doi.org/10.1103/3tn7-pqj4
We study the effect of subsystem decoherence to an extended cluster state which is a symmetry protected topological (SPT) phase. The model includes many subsystem symmetries. We report that subsystem decoherence induces local charge fluctuations, leading to a mixed SPT state in the unaffected subsystems. If we start from the extended cluster state, hierarchical mixed-state SPT phases emerge in response to step-by-step subsystem decoherences. These mixed-state SPT phases keep strong symmetries the symmetry of which is protecting symmetries for the initial cluster SPT. Moreover, these SPTs can be characterized by R'{e}nyi-2 string orders. Then, as the subsystems are progressively decohered, the hierarchy of mixed-state SPT phases terminates in a strong-to-weak spontaneous symmetry breaking (SWSSB) state on the final remaining subsystem, where a long-range entangled state appears, namely a glassy Greenberger-Horne-Zeilinger (GHZ) state. Our work demonstrates that decoherence is not merely a destructive process, but can induce and organize series of nontrivial mixed-states. This reveals a systematic route from mixed-state SPT order to SWSSB with the glassy GHZ-type long-range entanglement.
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