The Physics of Quantum Hall Systems and Edge States

Introduction

Quantum Hall systems are a class of quantum mechanical systems that exhibit the Quantum Hall Effect (QHE). The QHE is a quantum mechanical version of the classical Hall effect, which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. The physics of these systems is rich and complex, involving concepts from condensed matter physics, quantum mechanics, and topology.

Quantum Hall Effect

The Quantum Hall Effect (QHE) is a quantum mechanical phenomenon observed in two-dimensional electron systems. When these systems are subjected to low temperatures and a strong perpendicular magnetic field, they exhibit quantized Hall resistance and vanishing longitudinal resistance. This quantization is remarkably precise, independent of the material properties, and only depends on fundamental constants, the electron charge, and Planck's constant. This makes the QHE a standard for electrical resistance.

Two-Dimensional Electron Systems

The QHE is observed in two-dimensional electron systems (2DES). These are systems where the motion of electrons is restricted to two dimensions. Examples of 2DES include the interface between two different semiconductors, known as a heterostructure, and the surface of liquid helium. The restriction to two dimensions leads to unique electronic properties, such as the formation of a quantum well, which is key to the observation of the QHE.

Landau Levels

In a 2DES subjected to a magnetic field, the electrons move in circular orbits known as cyclotron orbits. The energy of these orbits is quantized into discrete levels known as Landau levels, named after the physicist Lev Davidovich Landau. The existence of Landau levels is a direct consequence of the quantum mechanical nature of electrons and is crucial to the understanding of the QHE.

Edge States

One of the most intriguing aspects of Quantum Hall systems are the edge states. These are states that are localized at the edges of the sample and travel in one direction only. The existence of these edge states is a direct consequence of the topological nature of the QHE and is responsible for the vanishing longitudinal resistance observed in the QHE.

Topological Properties

The QHE is a topological phenomenon. This means that it is robust against small changes in the system, such as impurities or disorder. This topological robustness is a direct consequence of the existence of edge states and is key to the precision of the QHE as a resistance standard.

Applications

The QHE has found applications in precision measurements and quantum computing. The precision of the QHE makes it an ideal standard for electrical resistance. Moreover, the topological nature of the QHE has been exploited in the development of topological quantum computers, which are expected to be more robust against errors.