Recent groundbreaking experiments on twisted semiconductor bilayers [1-6] and hBN-rhombohedral pentalayer graphene [7]  moiré superlattices demonstrate the fractional quantum anomalous Hall (FQAH) effect for the first time, ushering in a new era of topological quantum many-body physics. Here we report on several theoretical and numerical studies on the many-body phase diagram of tMoTe2 in the filling range 0 < n < 1 where n is the number of holes per moiré unit cell. We show that full spin/valley polarization occurs throughout most of this filling window and across a broad range of twist angles. Near a magic angle, an abundance of lowest Landau level (LLL) -like phases appear, including FQAH states and anomalous composite Fermi liquids. Away from this magic angle, some LLL-like features of the phase diagram persist. In contrast, others are replaced by competitors, namely charge density waves near n=0 and anomalous Hall Fermi liquids near n=1. Remarkably, we find that the anomalous composite Fermi liquid state appearing at n=1/2 is the LLL-like state most robust against deviation from the magic angle. Our study thoroughly establishes the close connection between the many-body phase diagrams of twisted MoTe2 and the LLL near the magic angle. Crucially, we go beyond identifying the various LLL-like phases that may appear under fine-tuned circumstances, developing an understanding of why some are more robust in experiments than others. We expect several guiding principles learned through our studies of tMoTe2 to apply to FQAH systems more generally.

[1] Anderson, Eric, et al. Science (2023)

[2] Cai, Jiaqi, et al. Nature (2023)

[3] Zeng, Yihang, et al. Nature (2023)

[4] Park, Heonjoon, et al. Nature (2023)

[5] Xu, Fan, et al. Physical Review X 13.3 (2023)

[6] Foutty, Benjamin A., et al. arXiv:2304.09808 (2023)

[7]  Lu, Zhengguang, et al. arXiv:2309.17436 (2023)