Topologically nontrivial and trivial flat bands via weak and strong interlayer coupling in twisted bilayer honeycomb optical lattices for ultracold atoms

In recent years, flat electronic bands in twisted bilayer graphene (TBG) have attracted significant attention due to their intriguing topological properties, extremely slow electron velocities, and enhanced density of states. Extending twisted bilayer systems to new configurations is highly desirable, as it offers promising opportunities to explore flat bands beyond TBG. Here, we study both topological and trivial flat bands in a twisted bilayer honeycomb lattice for ultracold atoms and present the evolution of the flat bands with different interlayer coupling strength (ICS). Our results demonstrate that an isolated topological flat band can emerge at the Dirac-point energy for a specific value of weak ICS, referred to as the ``critical coupling.'' This occurs over a wide range of twist angles, surpassing the limits of the magic angle in TBG systems. When the ICS is slightly increased beyond the critical coupling value, the topological flat band exhibits degenerate band crossings with both the upper and lower adjacent bands at the high-symmetry ${\mathrm{\ensuremath{\Gamma}}}_{s}$ point. As the ICS is further increased into the strong coupling regime, trivial flat bands arise around Dirac-point energy. Meanwhile, more trivial flat bands appear, extending from the lowest to higher energy bands, and remain flat as the ICS increases. The topological properties of the flat bands are studied through the winding pattern of the Wilson loop spectrum. Our research provides deeper insights into the formation of flat bands in ultracold atoms with highly controllable twisted bilayer optical lattices, and may contribute to the discovery of new strongly correlated states of matter.