Study on topological phase in one-dimensional momentum space lattice of ultracold atoms without chiral symmetry

Symmetry plays a crucial role in understanding topological phases in materials. In onedimensional systems, such as the Su-Schrieffer-Heeger (SSH) model, chiral symmetry was thought to ensure the quantization of the Zak phase and the nontrivial topological phases. However, our work demonstrates that the one-dimensional lattice system with broken chiral symmetry can still possess quantized Zak phase and nontrivial topological phases. Specifically, we use a Bose-Einstein condensate of 87Rb atoms in a momentum space lattice of ultracold atoms to effectively simulate a one-dimensional Zigzag model of 26 sites, which intrinsically breaks the chiral symmetry with additional next-nearest-neighbor coupling. To ensure the existence of the nontrivial topological phase, where the Zak phase can be measured from the time-averaged mean displacement during the system’s evolution, we need to preserve the inversion symmetry by modulating laser powers so that all next-nearest-neighbor coupling strengths are equal. Furthermore, by varying the ratio of nearest-neighbor coupling strengths, we observed a topological phase transition from a nontrivial to a trivial topological phase at the point where the ratio equals 1. Our work demonstrates that the ultracold atom system provides a controllable platform to study the symmetry and topological phases, with the potential to explore nonlinear topological phenomena by increasing the interactions among atoms. In addition, our system can be used to investigate other interesting topological phenomena with more complex models, such as critical phenomena at the phase transitions and flat band structures in the extended SSH models with long-range couplings. By controlling the coupling strengths, we can also explore the influence of different symmetries on the topological properties of extended SSH models in the future. Moreover, our platform enables the study of models with more energy bands, such as the Aharonov-Bohm caging model with a three-level structure, which exhibits peculiar flat band properties. This work provides opportunities to versatile studies in the realms of symmetry, topology, and their interplay in controllable quantum systems.