3D Quantum Hall Effect

Abstract

The integer quantum Hall effect (QHE) is the quantization of the Hall resistivity at integer plateau values, with corresponding zero longitudinal resistivity, which occurs at low temperatures and high magnetic fields. It is remarkable because it is the quantization of a bulk property of a material involving huge numbers of particles and an impure and disordered system. As such, it has been a topic of intense interest among physicists for decades and its study has led to breakthroughs in several areas of physics. It turns out that the QHE depends on special edge modes as well as topological properties of the system and that the QHE becomes apparent when the bulk of the system is an insulator and that it actually requires the presence of some disorder. Importantly, the QHE is a two-dimensional phenomenon. However, theorists have developed expected signatures for a 3D quantum hall effect and such an effect was recently experimentally observed for the first time. The experiment also observed unexpected plateau behavior in the 3D QHE. This thesis explains the fundamental basis of the 2D QHE and then goes on to explore one promising theoretical explanation for the emergence of a 3D QHE, namely the interaction of phonons and the 3D quantum Hall (QH) system to form charge density waves (CDWs). To explore this possibility, we numerically simulated the 3D QH system and used the method of gradient decent to optimize the overall energy while exploring the space of possible phonon modes under several different sets of parameters. We found a variety of interesting behaviors away from analytically tractable situations and when multiple Landau levels were occupied. Most importantly, we found two distinct phases occur when transitioning between occupying one and two Landau levels, including a metallic phase that could explain the observed plateaus in the 3D QHE.