Quantum Hall physics: Hierarchies and conformal field theory techniques

The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past 30 years, has generated many ground-breaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the 30 years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of non-Abelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.

Figure 1

Very high quality quantum Hall data showing many plateaus in the lowest Landau level. From [240].

Figure 2

Very high quality quantum Hall data showing plateaus in the second Landau level. Note that many of the plateaus labeled RIQHE (reentrant integer quantum Hall effect) are not fractional quantum Hall states but are believed to be some sort of charge density wave. From [191].

Figure 3

Observed states and fractal structure: quantum Hall hierarchy states in the lowest Landau and their relative stability. For each rational $\nu =p/q<1,q$ odd, there is a unique state and its stability increases monotonically with $1/q$. The crosses denote observed states in the region $2/7\le \nu \le 2/3$ ([240]). The horizontal line marks the extent in $\nu$ of the experiment and is a line of constant gap. This line provides a good (although not perfect) approximate boundary for the observed states. The inset shows the structure of hierarchy states: At each $\nu =p/q$, $q$ odd, there is a state with gap $\sim 1/q$ and quasiparticles with charge $\pm e/q$. When these condense, two sequences of states approaching $p/q$ with decreasing gap are obtained. From [22].