# Quantum Hall effects in novel 2D electron systems : nontrivial Fermi surface topology and quantum Hall ferromagnetism

## Abstract

In this thesis we discuss quantum Hall effects in bilayer graphene and other novel two-dimensional electron systems, focusing on the interplay between nontrivial Fermi surface topology and electron-electron interactions. In the first chapter I will give a brief introduction to some aspects of the quantum Hall effects. The second chapter discusses the physics in bilayer graphene in the absence of external magnetic fields. The first half discusses the band gap opening and trigonal warping effects in its bandstructure, and the second half focuses on the insulating ground state that results from electron-electron interactions. The third chapter discusses the single-particle Landau level structure in bilayer graphene. We will see that when both the band gap and trigonal warping effects are present, the highest Landau level in the valence band is three-fold degenerate at small magnetic fields. As the field increases, the three fold degeneracy is lifted and the Landau level structure gradually reduces to that in the absence of trigonal warping effects. At the end of the chapter we will demonstrate a formalism to map the momentum distribution of the single-particle Landau level structure. Such a mapping will give valuable information about the single-particle bandstructure. The fourth chapter deals with electron-electron interactions in the integer quantum Hall regime, where there is no fractional filling of the orbital degrees of freedom. In such a regime, the effect of electron-electron interactions often leads to spontaneous ordering of the internal degrees of freedom, such as spin, layer and valley. The first part of the chapter will establish the general formalism of Hartree-Fock theory in the quantum Hall regime, and then a specific theory for gapped bilayer graphene with trigonal warping effects is constructed. The resulting ground states are analyzed in the last part of the chapter.