Quantum groups and non-Abelian braiding in quantum Hall systems

Abstract

Wave functions describing quasiholes and electrons in nonabelian quantum Hall states are well known to correspond to conformal blocks of certain coset conformal field theories. In this paper we explicitly analyse the algebraic structure underlying the braiding properties of these conformal blocks. We treat the electrons and the quasihole excitations as localised particles carrying charges related to a quantum group that is determined explicitly for the cases of interest. The quantum group description naturally allows one to analyse the braid group representations carried by the multi-particle wave functions. As an application, we construct the nonabelian braid group representations which govern the exchange of quasiholes in the fractional quantum Hall effect states that have been proposed by N. Read and E. Rezayi [1], recovering the results of C. Nayak and F. Wilczek [2] for the Pfaffian state as a special case.

Realizing non-Abelian statistics in time-reversal-invariant systems

“…taking the θ → ∞ limit of the S matrix) gives the braid matrix of particles associated with SU (2) k Chern-Simons theory. This is also the braiding of the quasiholes in the Read-Rezayi states in the fractional quantum Hall effect 20,41,48 . It is important to note that this representation, Eq.…”

“…One can define an action of U q (sl 2 ) on the space of states V ⊗N which commutes with the e i ; see Ref. 41 for an extensive discussion of quantum groups in the context of non-Abelian statistics (q there is q 2 here). In particular, to relate the two algebras, first note that e i /d is a projector.…”

“…What this means is that at each vertex, configurations appearing in the ground state must obey the corresponding fusion rules of U q (sl 2 ). 34,41,66,67,68 . For example, three links, in states corresponding to representations r, s and t of U q (sl 2 ), touch each vertex of the honeycomb lattice.…”

“…To count the "number" of such kink configurations for large N , one applies the same procedure as above, and obtains 2 cos N (π/(k + 2)). In the Read-Rezayi generalizations of the Moore-Read theory of the fractional quantum Hall effect 20 , the statistics and the number of states is the same 41 .…”

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

“…The Jones-Wenzl idempotent which has index n means the projection to spin-n/2 space, and the arc labelled by n corresponds to the Wilson line of quasi-particle with spin-n/2; n : spin-n/2 quasi-particles Then the trivalent vertex (2.23) denotes the usual fusion rule; quasi-particles, φ a and φ b , with spina/2 and spin-b/2 fuse to spin-c/2 quasi-particle φ c . One sees that the admissible condition (2.24) of the SU(2) K CS theory has a correspondence with the fusion channel (2.2) [56]. In general the fusion rule is written as…”