We review analytical and numerical studies of correlated insulating states in twisted bilayer graphene, focusing on real-space lattice models constructions and their unbiased quantum many-body solutions. We show that by constructing localized Wannier states for the narrow bands, the projected Coulomb interactions can be approximated by interactions of cluster charges with assisted nearest neighbor hopping term. With the interaction part only, the Hamiltonian is $SU(4)$ symmetric considering both spin and valley degrees of freedom. In the strong coupling limit where the kinetic terms are neglected, the ground states are found to be in the $SU(4)$ manifold with degeneracy. The kinetic terms, treated as perturbation, break this large $SU(4)$ symmetry and propel the appearance of intervalley coherent state, quantum topological insulators and other symmetry-breaking insulating states. We first present the theoretical analysis of moire lattice model construction and then show how to solve the model with large-scale quantum Monte Carlo simulations in an unbiased manner. We further provide potential directions such that from the real-space model construction and its quantum many-body solutions how the perplexing yet exciting experimental discoveries in the correlation physics of twisted bilayer graphene can be gradually understood. This review will be helpful for the readers to grasp the fast growing field of the model study of twisted bilayer graphene.

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扭曲双层石墨烯中的相关绝缘相

我们回顾了扭曲双层石墨烯中相关绝缘状态的分析和数值研究，重点关注实空间晶格模型构造及其无偏量子多体解。我们表明，通过为窄带构建局部万尼尔状态，可以通过簇电荷与辅助最近邻跳跃项的相互作用来近似预测库仑相互作用。仅在相互作用部分，考虑到自旋和谷自由度，哈密顿量是 $SU(4)$ 对称的。在忽略动力学项的强耦合极限中，发现基态处于具有简并性的 $SU(4)$ 流形中。被视为扰动的动力学项打破了这种大的 $SU(4)$ 对称性并推动了间隔相干态的出现，量子拓扑绝缘体和其他破坏对称性的绝缘状态。我们首先介绍了莫尔晶格模型构建的理论分析，然后展示了如何以无偏的方式使用大规模量子蒙特卡罗模拟来求解模型。我们进一步提供了潜在的方向，以便从实空间模型构建及其量子多体解决方案如何逐步理解扭曲双层石墨烯相关物理学中令人困惑但令人兴奋的实验发现。这篇综述将有助于读者掌握扭曲双层石墨烯模型研究的快速发展领域。我们首先介绍了莫尔晶格模型构建的理论分析，然后展示了如何以无偏的方式使用大规模量子蒙特卡罗模拟来求解模型。我们进一步提供了潜在的方向，以便从实空间模型构建及其量子多体解决方案如何逐步理解扭曲双层石墨烯相关物理学中令人困惑但令人兴奋的实验发现。这篇综述将有助于读者掌握扭曲双层石墨烯模型研究的快速发展领域。我们首先介绍了莫尔晶格模型构建的理论分析，然后展示了如何以无偏的方式使用大规模量子蒙特卡罗模拟来求解模型。我们进一步提供了潜在的方向，以便从实空间模型构建及其量子多体解决方案如何逐步理解扭曲双层石墨烯相关物理学中令人困惑但令人兴奋的实验发现。这篇综述将有助于读者掌握扭曲双层石墨烯模型研究的快速发展领域。