Topological Phases in a Dipolar Bose-Hubbard Quantum Simulator Part II: Mixed State Order and the Haldane Insulator

Using an erbium quantum gas microscope, we further experimentally investigate two intriguing aspects of the crystalline symmetry-protected topological (CSPT) phases we observe. First, we demonstrate that these phases remain robust under disorder averaging, even though individual disorder realizations break the protecting crystalline symmetry. This establishes the CSPT phase as an example of mixed-state quantum order. To verify this experimentally, we introduce a controlled, programmable random chemical potential disorder, which disrupts order parameters at the individual realization level. However, averaging over multiple disorder realizations restores the distinct phase structure, confirming the topological phase’s robustness. Finally, we establish a connection between the Haldane insulator, a well-known symmetry-protected topological (SPT) phase, and the CSPT phases. In particular, we show that the CSPT transition interpolates to a transition between a Mott insulator to Haldane insulator after removing the alternating chemical potential. We observe order parameters consistent with this transition, providing the first experimental evidence of a tunable, interacting SPT phase in an analog setting.