Realization of the Symmetry-Protected Haldane phase in Fermi-Hubbard Ladder Systems*

The concept of topology has strongly changed our understanding of quantum phases in many-body systems. The antiferromagnetic spin-1 Haldane chain has been used as the model for introducing these unexpected effects. The chain has symmetry-protected fourfold degenerate edge states with spin-1/2 localized at the two edges, while inside the chain, an excitation gap with a hidden long-range topological order prevails. Here, we introduce the experimental realization of the topological Haldane phase in Fermi-Hubbard ladders using ultracold-atom quantum simulators. This bases on the AKLT model in which a spin-1 particle is built-up out of two spin-1/2 particles. From the results, we directly observe both edge and bulk characteristics by harvesting our unique detection scheme with full spin and density resolution that allows us to construct string order parameters. By reducing the Hubbard interaction strength of the system far from the Heisenberg regime, we observe the robustness of the phase against the density fluctuations.