After more than four years of intense research, the nature of the ground states of twisted bilayer graphene (tBLG) still remains a mystery. In this talk, I will discuss numerical and experimental techniques that can be employed to study ground states. 

In the first part of the talk, I will discuss the development of the density matrix renormalization group method for twisted bilayer graphene. Owing to the complexity of the Hamiltonian, this required a novel method to represent the Hamiltonian in a compressed format using so-called "MPO compression" technique. This technique allowed us to identify various ground states of tBLG, such as quantum anomalous Hall state, nematic semimetal, Kramers-intervalley coherent state, and incommensurate Kekule spiral.

In the second part of the talk, I will discuss how we can use scanning tunneling microscopy (STM) to detect symmetry breaking in tBLG. We observe, by analogy with the physics of the zeroth Landau levels of monolayer graphene, that symmetry breaking signal directly shows up as atomic scale variation in local density of states (LDOS). We show that almost all of the competing candidate states have distinct signatures in LDOS observable in LDOS, some of which only shows up after applying some magnetic field.