We propose a one-dimensional optical lattice model to simulate and explore two-dimensional topological phases with ultracold atoms, considering the phases of the hopping strengths as an extra dimension. It is shown that the model exhibits nontrivial phases, and corresponding two chiral-edge states. Moreover, we demonstrate the connections between changes in the topological invariants and the Dirac points. Furthermore, the topological order detected by the particle pumping approach in cold atoms is also investigated. The results obtained here provide a feasible and flexible method of simulating and exploring high-dimensional topological phases in low-dimension systems via the controllable phase of the hopping strength.

我们提出了一个一维光学晶格模型来模拟和探索具有超冷原子的二维拓扑相，将跳跃强度的相视为一个额外的维度。结果表明，该模型表现出非平凡的阶段，以及相应的两个手征边缘状态。此外，我们证明了拓扑不变量的变化与狄拉克点之间的联系。此外，还研究了在冷原子中通过粒子泵浦方法检测到的拓扑顺序。此处获得的结果为通过跳跃强度的可控相位模拟和探索低维系统中的高维拓扑相位提供了一种可行且灵活的方法。