Quantum anomalous Hall phase in synthetic bilayers via twistless twistronics

We recently proposed quantum simulators of "twistronic-like" physics based on ultracold atoms and syntheticdimensions [Phys. Rev. Lett. 125, 030504 (2020)]. Conceptually, the scheme is based on the idea that aphysical monolayer optical lattice of desired geometry is upgraded to a synthetic bilayer system by identifyingthe internal states of the trapped atoms with synthetic spatial dimensions. The couplings between the internalstates, i.e. between sites on the two layers, can be exquisitely controlled by laser induced Raman transitions.By spatially modulating the interlayer coupling, Moir\'e-like patterns can be directly imprinted on the latticewithout the need of a physical twist of the layers. This scheme leads practically to a uniform pattern across thelattice with the added advantage of widely tunable interlayer coupling strengths. The latter feature facilitates theengineering of flat bands at larger "magic" angles, or more directly, for smaller unit cells than in conventionaltwisted materials. In this paper we extend these ideas and demonstrate that our system exhibits topologicalband structures under appropriate conditions. To achieve non-trivial band topology we consider imanaginarynext-to-nearest neighbor tunnelings that drive the system into a quantum anomalous Hall phase. In particular,we focus on three groups of bands, whose their Chern numbers triplet can be associated to a trivial insulator(0,0,0), a standard non-trivial (-1,0,1) and a non-standard non-trivial (-1,1,0). We identify regimes of parameterswhere these three situations occur. We show the presence of an anomalous Hall phase and the appearance oftopological edge states. Our works open the path for experiments on topological effects in twistronics without atwist