Interacting Electrons in Twisted Graphene Moiré Lattices

Abstract

By stacking two layers of 2-dimensional materials on top of each other, a superlattice will emerge when the two sheets are twisted by a small angle or have a lattice mismatch. This phenomenon is known as ``moir\'e pattern''. Twisted bilayer graphene (TBG) is one of the most important examples of materials with moir\'e patterns. Theoretical studies predicted the existence of flat bands when it is twisted by the magic angle theta=1.05 degrees. Magic angle TBG was first realized experimentally in 2018, and correlated insulating and superconducting phases were observed. The discovery of unconventional correlated phenomena triggered abundant theoretical researches about the electron interactions in TBG and other moir\'e materials. This thesis is focusing on my theoretical and numerical studies of the interacting electrons in the twisted multilayer graphene systems. In Chapter 1, I briefly review the experimental discoveries of TBG and the derivation of the well-known continuum non-interacting Hamiltonian introduced by R. Bistritzer and A. Macdonald. Symmetries of this Hamiltonian are also discussed in this chapter. Chapter 2 is based on the work with Zhida Song, Biao Lian and B. Andrei Bernevig [1]. It focuses on the relationship between the flat band topology and an important transport parameter ``superfluid weight'', which measures the stiffness of superconducting order parameter phase correlation. Chapter 3 is based on the work with Aditya Cowsik, Zhida Song, Biao Lian, B. Andrei Bernevig and Nicolas Regnault [2]. By projecting Coulomb interaction into the low-energy degrees of freedom, it becomes possible to perform unbiased exact diagonalization calculation of interacting TBG on a small lattice. It unveils the phase diagram and several ground state wavefunctions of the correlated insulating phases. Chapter 4 is based on the work with Nicolas Regnault, Dumitru Calugaru, B. Andrei Bernevig and Biao Lian [3]. It is committed to study the interacting electrons in twisted symmetric trilayer graphene (TSTG) via mean field approximation. It uncovers the phase diagrams of TSTG at integer fillings, and the connection with correlated states in TBG. Results in Chapters 3 and 4 have been presented at APS March Meetings in 2021 and 2022.