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We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin–orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin–orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.

We study the extended Kitaev chain with both nearest-neighbor and next-nearest-neighbor hopping terms and find the model system exhibiting nontrivial phases, which can be characterized by a nonzero Berry phase and winding number when the system is in a pure state. While in a mixed state, we investigate the robustness of the topological Uhlmann phases and show how it responses to the presence of next-nearest-neighbor hopping terms. Furthermore, we analyse the complicated behavior of the Uhlmann phase of the extended Kitaev chain at finite temperature as k moves along the Brillouin zone, and we think this may serve as a topological indicator for mixed states in condensed matter systems.