Particle-hole symmetry in the anomalous quantum Hall effect

Abstract

This paper explores the uses of particle-hole symmetry in the study of the anomalous quantum Hall effect. A rigorous algorithm is presented for generating the particle-hole dual of any state. This is used to derive Laughlin's quasihole state from first principles and to show that this state is exact in the limit $\nu \to 1$, where $\nu$ is the Landau-level filling factor. It is also rigorously demonstrated that the creation of $m$ quasiholes in Laughlin's state with $\nu =\frac{1}{m}$ is precisely equivalent to creation of one true hole. The charge-conjugation procedure is also generalized to obtain an algorithm for the generation of a hierarchy of states of arbitrary rational filling factors.