Quasielectrons in Abelian and non-Abelian Quantum Hall States
Abstract: Strongly correlated electron systems continue to attract a lot of interest. Especially two-dimensional electron systems have shown many surprising behaviours. A fascinating example is the quantum Hall effect, which arises when electrons are confined to two dimensions, subjected to a very large magnetic field, and cooled to very low temperatures. Under these conditions, the electrons form new states of matter - the strongly correlated quantum Hall liquids. A hallmark of these quantum liquids is the precise quantization of the Hall conductance, but in recent years more attention has been focused on their exotic excitations. These have fractional electric charge, and fractional exchange statistics. The latter implies that the wave function is multiplied by a phase factor containing a fractional phase when two quasiparticles are moved around each other. Thus, they are neither bosons nor fermions, but so-called anyons. In recent years, there has accumulated theoretical and experimental evidence for some of the quantum Hall liquids having even more exotic excitations with not only fractional but non-Abelian exchange statistics. The quasiparticles of such systems could be used to build topologically protected quantum bits, which are much more robust than presently available quantum bits; they would be the ideal building blocks of a quantum computer. We use conformal field theory to describe the quantum Hall liquids, as well as their excitations. In particular, we represent the electrons and quasiparticles by conformal field theory operators. Even though the operator description for quasiholes is very well understood, it was for a long time unclear how to describe their antiparticles, the quasielectrons. We found an operator that describes quasielectron excitations correctly and shares many of the useful properties of the corresponding quasihole operator. For instance, many of the topological properties of the particles are manifest in the operator. This not only adds a missing piece to the quantum Hall puzzle, but it also opens up new and exciting possibilities. For instance, we were able to extend this construction to the non-Abelian states. A highly non-trivial application of our approach is the condensation of non-Abelian quasielectrons, which yields new non-Abelian quantum Hall states with non-Abelian properties that differ from those of their parent states.
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