  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 terms. 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 moir\'e 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.
