States of matter are conventionally classified according to broken
symmetries. Topologically ordered phases fall outside of this paradigm:
with no local order parameter, they nevertheless have many peculiar
properties setting them apart from disordered phases. In 2D, such
phases may support anyons - quasiparticles that are neither bosons
nor fermions. Moreover, anyons with non-Abelian statistics are
expected to occur in certain fractional quantum Hall systems as well
as materials supporting topological superconductivity (intrinsic or
induced).
In this talk, I will discuss various approaches designed to
detect such exotic statistics. I will also describe our recent
proposal for novel physical systems supporting unusual non-Abelian
anyons. I will also mention potential applications of such systems for
topological quantum computation.