UC Berkeley

When two layers of graphene are put on top of each another with a relative twist, their lattice mismatch gives rise to a moir {e} pattern. When the twist angle is near 1.1 degrees, a so-called magic angle, the band structure of twisted bilayer graphene becomes extremely flat around the Fermi energy, enhancing the effect of electron-electron interaction. Remarkably, magic-angle twisted bilayer graphene (MATBG) becomes superconducting at low temperatures. The origin of this superconducting behavior remains elusive.

Curiously, the superconducting behavior is usually accompanied by correlated insulating behavior in nearby parameter regions in the phase diagram. These correlated insulators cannot be described by non-interacting band theory, hinting at the importance of electron-electron interaction.

In this thesis, we attempt to understand these correlated insulators by a combination of numerical and analytical techniques. On the numerical side, we will utilize the density-matrix renormalization group (DMRG) extensively to obtain the ground states of MATBG. Owing to the complexity of the problem, this required us to develop a non-trivial routine for encoding the Hamiltonian. We find that DMRG often reproduces the findings from Hartree-Fock simulations.On the analytical side, we aim to understand different symmetry-breaking states of MATBG. Based on symmetry analysis, we propose using scanning tunneling microscopy (STM) to distinguish between different candidate ground states, and test our analytical prediction using numerical simulation.

Taken together, this thesis represents a significant step toward understanding the correlated insulating behavior of MATBG.