Correlated Insulating Phases in the Twisted Bilayer Graphene

TL;DR

This paper reviews theoretical and numerical studies of correlated insulating states in twisted bilayer graphene, emphasizing real-space models, symmetry considerations, and quantum Monte Carlo solutions to understand experimental phenomena.

Contribution

It introduces a comprehensive real-space lattice model approach and unbiased quantum Monte Carlo methods to analyze correlated insulating phases in twisted bilayer graphene.

Findings

Localized Wannier states enable modeling of Coulomb interactions.

Strong coupling limit exhibits $SU(4)$ symmetric ground states.

Kinetic terms break symmetry, leading to various insulating states.

Abstract

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.