Theory of Quantum Hall States

Abstract

Electrons on the interface of two semiconductors can form a new state of matter — fractional quantum Hall (FQH) state — under strong magnetic fields. FQH states cannot be described by Landau symmetry breaking theory. These shatter the long-held belief that symmetry breaking theory describes all phases and phase transitions. As a result, a completely new theory is needed to describe FQH states, and this is the topic of this chapter. Many-electron systems in strong magnetic fields and resulting Landau level structures are studied. Laughlin's theory and the hierarchical theory for FQH effect are presented. The chapter then derives the low energy effective Chern–Simons theory for FQH states and discusses the resulting fractional charge and fractional statistics, as well as the K-matrix classification of Abelian FQH states. The theory of chiral gapless edge states is also introduced, where experimental predictions can be made.