Quantum Hall quasielectron operators in conformal field theory

TL;DR

This paper constructs a new conformal field theory operator for quasielectrons in quantum Hall states, enabling explicit wave functions and clarifying their statistics, thus advancing the theoretical understanding of these excitations.

Contribution

It introduces a novel quasielectron operator in CFT that reproduces known wave functions, predicts new ones, and clarifies the statistics of quasiparticles in hierarchical quantum Hall states.

Findings

Constructed a quasielectron operator satisfying symmetry, braiding, and fusion constraints.

Provided explicit wave functions for Moore-Read Pfaffian state quasielectrons.

Expressed composite fermion wave functions in hierarchical form, resolving a longstanding controversy.

Abstract

In the conformal field theory (CFT) approach to the quantum Hall effect, the multi-electron wave functions are expressed as correlation functions in certain rational CFTs. While this approach has led to a well-understood description of the fractionally charged quasihole excitations, the quasielectrons have turned out to be much harder to handle. In particular, forming quasielectron states requires non-local operators, in sharp contrast to quasiholes that can be created by local chiral vertex operators. In both cases, the operators are strongly constrained by general requirements of symmetry, braiding and fusion. Here we construct a quasielectron operator satisfying these demands and show that it reproduces known good quasiparticle wave functions, as well as predicts new ones. In particular we propose explicit wave functions for quasielectron excitations of the Moore-Read Pfaffian state. Further, this operator allows us to explicitly express the composite fermion wave functions in the positive Jain series in hierarchical form, thus settling a longtime controversy. We also critically discuss the status of the fractional statistics of quasiparticles in the Abelian hierarchical quantum Hall states, and argue that our construction of localized quasielectron states sheds new light on their statistics. At the technical level we introduce a generalized normal ordering, that allows us to "fuse" an electron operator with the inverse of an hole operator, and also an alternative approach to the background charge needed to neutralize CFT correlators. As a result we get a fully holomorphic CFT representation of a large set of quantum Hall wave functions.