Obstructed superfluid in the attractive Qi-Wu-Zhang-Hubbard model

Motivated by the recent realization of the Qi-Wu-Zhang (QWZ) model for the minimal spin-orbit-coupled quantum anomalous Hall phase in ultracold fermions, we introduce the on-site attractive interaction and map out the topological phase diagram for the Qi-Wu-Zhang-Hubbard (QWZH) model at the mean-field level. We show that, in addition to the conventional chiral topological superfluids (TSs) with nonzero Chern number, there is also an obstructed superfluid (OSF) with zero Chern number in the extended region of phase space. This phase is characterized by gapless edge states completely detaching from the bulk energy bands and two ${Z}_{2}$ numbers. We find the gapless edge states in OSF are protected by particle-hole symmetry and robust for the disorder. The appearance of OSF demonstrates that it is not sufficient to label TSs in the two-dimensional D class only by Chern numbers. These different topological phases in the attractive QWZH model can be detected experimentally by spin-resolved time-of-flight imaging. We also plot the phase diagram of the attractive QWZH model with one-dimensional spin-orbit coupling and find a series of TSs with gapless Dirac points. Considering the successful realization of the QWZ model and the tremendous progress in manipulating cold atom systems, our work provides a possible route to realize OSF in cold fermionic systems in the future.