Topological phase transitions and mixed-state order in a Hubbard quantum simulator

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Topological phase transitions separate many-body phases that are locally indistinguishable yet globally distinct. Here we show that such a transition can be identified between one-dimensional crystalline-symmetry-protected topological phases using a quantum simulator of interacting erbium atoms in an optical lattice. We detect the critical point through non-local string order parameters and reveal its connection to the transition predicted between Mott and Haldane insulators. We also show that stacking two identical systems eliminates the transition, consistent with the predicted group structure and the invertibility of symmetry-protected topological phases. Finally, introducing symmetry-breaking disorder removes the transition, whereas disorder averaging restores it. The adjacent phases, therefore, realize a form of mixed-state quantum order in which the criticality between them depends on the observer’s information. Our results show how topology and information shape quantum phase transitions in programmable quantum matter.

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The experimental data supporting the findings of this study are available via Zenodo at https://doi.org/10.5281/zenodo.15627984 (ref. 90 ).

The analysis code supporting the findings of this study are available via Zenodo at https://doi.org/10.5281/zenodo.15627984 (ref. 90 ).

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