Topological Phases in a Dipolar Bose-Hubbard Quantum Simulator Part I: Crystalline SPT*

Topological phases of matter extend beyond the Landau paradigm of symmetry breaking, with certain phases in quantum many-body systems being protected by symmetries. We present the first experimental realization of crystalline symmetry-protected topological (CSPT) phases in the dipolar Bose-Hubbard model, using magnetic erbium atoms in optical lattices. These systems feature long-range interactions and on-site interactions tunable via magnetic Fano-Feshbach resonances. By introducing an alternating chemical potential, we observe quantum phase transitions between two CSPT phases, which are characterized using site-resolved imaging that directly measures the atom number per site without parity projection. The parity order parameter (POP) effectively distinguishes these phases, while local measurements fail to do so, underscoring their topological nature. To showcase the $\mathbb{Z}_2$ topological property, we experimentally realize an even number of coupled SPT chains and observe the trivialization of the quantum phase transition between them for the first time.