The k=3 Read-Rezayi (RR) parafermion quantum Hall state hosts non-Abelian
excitations which provide a platform for the universal topological quantum
computation. Although the RR state may be realized at the filling factor
\nu=12/5 in GaAs-based two-dimensional electron systems, the corresponding
quantum Hall state is weak and at present nearly impossible to study
experimentally. Here we argue that the RR state can alternatively be realized
in a class of chiral materials with massless and massive Dirac-like band
structure. This family of materials encompasses monolayer and bilayer graphene,
as well as topological insulators. We show that, compared to GaAs, these
systems provide several important advantages in realizing and studying the RR
state. Most importantly, the effective interactions can be tuned {\it in situ}
by varying the external magnetic field, and by designing the dielectric
environment of the sample. This tunability enables the realization of RR state
with controllable energy gaps in different Landau levels. It also allows one to
probe the quantum phase transitions to other compressible and incompressible
phases.Comment: 12 pages, 5 figures; to appear in New Journal of Physics, Focus on
Topological Quantum Computatio
