Using comprehensive exact diagonalization calculations on $θ\approx 3.7 ^{\circ}$ twisted bilayer MoTe$_2$ ($t$MoTe$_2$), as well as idealized Landau level models also relevant for lower $θ$, we extract general principles for engineering fractional topological insulators (FTIs) in realistic situations. First, in a Landau level setup at $ν=1/3+1/3$, we investigate what features of the interaction destroy an FTI. For both pseudopotential interactions and realistic screened Coulomb interactions, we find that sufficient suppression of the short-range repulsion is needed for stabilizing an FTI. We then study $θ\approx 3.7 ^{\circ}$ $t$MoTe$_2$ with realistic band-mixing and anisotropic non-local dielectric screening. Our finite-size calculations only find an FTI phase at $ν=-4/3$ in the presence of a significant additional short-range attraction $g$ that acts to counter the Coulomb repulsion at short distances. We discuss how further finite-size drifts, dielectric engineering, Landau level character, and band-mixing effects may reduce the required value of $g$ closer towards the experimentally relevant conditions of $t$MoTe$_2$. Projective calculations into the $n=1$ Landau level, which resembles the second valence band of $θ\simeq 2.1^\circ$ $t$MoTe$_2$, do not yield FTIs for any $g$, suggesting that FTIs at low-angle $t$MoTe$_2$ for $ν=-8/3$ and $-10/3$ may be unlikely. While our study highlights the challenges, at least for the fillings considered, to obtaining an FTI with transport plateaus, even in large-angle $t$MoTe$_2$ where fractional Chern insulators are experimentally established, we also provide potential sample-engineering routes to improve the stability of FTI phases.
Pressure-Tunable Generalized Wigner Crystal and Fractional Chern Insulator in twisted MoTe$_2$
Beijing Institute of Technology（北京理工大学）；Beijing Key Laboratory of Quantum Matter State Control and Ultra-Precision Measurement Technology（北京量子物质态调控与超精密测量技术重点实验室）
Fractional Chern insulators in alternating twisted multilayer MoTe$_{2}$
Huazhong University of Science and Technology（华中科技大学）；Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology（华中科技大学武汉国家高能磁场中心）；Hubei Fundamental Research Center for Physics（湖北省物理基础研究中心）
Moiré-modulated $Γ$ valley in twisted bilayer and twisted double-bilayer MoTe$_2$
Tsinghua University（清华大学）；Beijing Tsinghua Institute for Frontier Interdisciplinary Innovation（北京清华交叉信息研究院）；Shanghai Jiao Tong University（上海交通大学）；National Institute for Materials Science（国立研究开发法人物质·材料研究机构）；Synchrotron SOLEIL（SOLEIL同步辐射中心）；Diamond Light Source Ltd（钻石光源有限公司）；Institute for Advanced Study, Tsinghua University（清华大学高等研究中心）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）
Van Hove Singularity-Driven Topological Magnetism in Twisted MoTe2
University of Washington（华盛顿大学）；Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）；University of California - Santa Barbara（加利福尼亚大学圣巴巴拉分校）；National Institute for Materials Science（国立材料研究所）
Intervalley Band Crossing and Transition of Fractional Chern Insulators in Floquet Twisted Bilayer MoTe$_2$
Zhejiang Institute of Modern Physics, Zhejiang University（浙江大学现代物理研究所）；Zhejiang Key Laboratory of Micro-Nano Quantum Chips and Quantum Control, School of Physics, Zhejiang University（浙江大学物理学院微纳量子芯片与量子控制浙江省重点实验室）
Chiral superconductivity from spin polarized Chern band in twisted MoTe$_2$
University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）；Tsinghua University（清华大学）；Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Shanghai Jiao Tong University（上海交通大学）；Center for Integrated Nanotechnologies (CINT)（综合纳米技术中心）；Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校闵H. Kao电气与计算机工程系）
Large scale theoretical investigation of the phase diagram of twisted bilayer MoTe$_2$ at fractional fillings: agreements and contradictions with current experiments
Donostia International Physics Center（多诺西亚国际物理中心）；Department of Physics and Quantum Theory Project, University of Florida（佛罗里达大学物理系与量子理论项目）；Department of Physics, Princeton University（普林斯顿大学物理系）；Geballe Laboratory for Advanced Materials, Departments of Applied Physics, Stanford University（斯坦福大学应用物理系吉贝尔先进材料实验室）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯克，巴斯克科学基金会）；Center for Computational Quantum Physics, Flatiron Institute（平顿研究所计算量子物理中心）；Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité（巴黎高等师范学校物理实验室，ENS，PSL大学，法国国家科学研究中心，索邦大学，巴黎狄德罗大学，索邦巴黎城）
Non-monotonic band flattening near the magic angle of twisted bilayer MoTe$_2$
Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory（斯坦福材料能源科学研究所，SLAC国家加速器实验室）；Department of Physics, University of Washington（华盛顿大学物理系）；Department of Physics, Boston College（波士顿学院物理系）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）；Department of Applied Physics, Stanford University（斯坦福大学应用物理系）；Research Center for Materials Nanoarchitectonics, National Institute for Materials Science（物质材料研究机构材料纳米结构研究中心）；Research Center for Electronic and Optical Materials, National Institute for Materials Science（物质材料研究机构电子光学材料研究中心）；Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory（SLAC国家加速器实验室斯坦福同步辐射光源）；Advanced Light Source, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室先进光源）；Department of Materials Science and Engineering, Stanford University（斯坦福大学材料科学与工程系）；Department of Materials Science and Engineering, University of Washington（华盛顿大学材料科学与工程系）；Department of Physics, Stanford University（斯坦福大学物理系）；Geballe Laboratory for Advanced Materials, Stanford University（斯坦福大学吉贝尔高级材料实验室）
Superconductivity Proximate to Non-Abelian Fractional Spin Hall Insulator in Twisted Bilayer MoTe$_2$
Pohang University of Science and Technology（浦项科技大学）；Ulsan National Institute of Science and Technology（蔚山科学技术院）；Daegu Gyeongbuk Institute of Science and Technology (DGIST)（大邱庆北科学技术院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Institute for Basic Science（基础科学研究院）
Fractional Chern Insulator and Quantum Anomalous Hall Crystal in Twisted MoTe$_2$
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；National High Magnetic Field Lab（国家强磁场实验室）
Electric-field-tuned consecutive topological phase transitions between distinct correlated insulators in moire MoTe2/WSe2 heterobilayer
State Key Laboratory of Micro-nano Engineering Science, Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院，微纳工程科学国家重点实验室，人工结构与控制教育部重点实验室）；School of Applied and Engineering Physics, Cornell University（康奈尔大学应用与工程物理学院）；Departments of Physics and Applied Physics, Stanford University（斯坦福大学物理与应用物理系）；National Institute for Materials Science（日本物质材料研究机构）；SLAC National Accelerator Laboratory（斯坦福直线加速器国家实验室）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Laboratory of Atomic and Solid State Physics, Cornell University（康奈尔大学原子与固态物理实验室）；Kavli Institute at Cornell for Nanoscale Science（康奈尔大学卡弗里纳米科学研究所）；Max Planck Institute for the Structure and Dynamics of Matter（德国马普物质结构与动力学研究所）
School of Physics and Astronomy, University of Minnesota（明尼苏达大学物理与天文学院）；William I. Fine Theoretical Physics Institute, University of Minnesota（明尼苏达大学威廉·I·芬恩理论物理研究所）
Interplay between topology and electron-electron interactions in the moiré MoTe$_{\mathrm{2}}$/WSe$_{\mathrm{2}}$ heterobilayer
Academic Centre for Materials and Nanotechnology, AGH University of Krakow（AGH科技大学材料与纳米技术学术中心）；Department of Quantum Matter Physics, University of Geneva（日内瓦大学量子物质物理系）
From fractional Chern insulators to topological electronic crystals in moiré MoTe2: quantum geometry tuning via remote layer
Shanghai Jiao Tong University（上海交通大学）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交大凌波研究院）；Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）；National Institute for Materials Science（国立材料研究所）；Hefei National Laboratory（合肥国家实验室）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；University of Tennessee（田纳西大学）
Characterization of fractional Chern insulator quasiparticles in twisted homobilayer MoTe$_2$
Zhejiang Institute of Modern Physics, Zhejiang University（浙江大学现代物理研究所）；School of Physics, Zhejiang University（浙江大学物理学院）；School of Physics and Technology, Wuhan University（武汉大学物理科学与技术学院）；Wuhan Institute of Quantum Technology（武汉量子技术研究院）
Candidate for a Fractional Topological Insulator in Twisted MoTe2
Columbia University（哥伦比亚大学）；University of Washington（华盛顿大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；National Institute for Materials Science（国立材料研究所）；Flatiron Institute（平顿研究所）；Princeton University（普林斯顿大学）；PSL University（巴黎文理研究大学）
