Topological Properties of Quantum States of Condensed Matter:
Some Recent Surprises

F.D.M. Haldane, Princeton University

It is now over twenty-five years since both the quantum Hall effect and
the importance of the Berry phase in quantum mechanics were discovered,
but they continue to inspire new developments in our understanding of
the possible quantum states of condensed matter systems. 

In the last few years, the old problem of the anomalous Hall effect in
ferromagnets has been revisited, and a modern understanding in terms of
Berry phases developed. A new classification of band insulators into
topologically-trivial and non-trivial classes has shown that the
two-dimensional surfaces of  non-trivial insulators support exotic
metallic states with an intrinsic "spin-Hall" effect that are
unaffected by the usual electron localization effects that destroy
conductivity in standard 2D metals, and it seems that these states have
already been found on the surface of Bi-Sb alloys.   

The properties of graphene sheets and their electronic "Dirac points" 
also provide examples of the importance of Berry phases and topology, and the
properties of the Dirac point in the presence of broken inversion or
time-reversal symmetry, or spin-orbit coupling, is a paradigm for many
of these phenomena: it has even been translated from electronic to
photonic states,  where it has suggested the possibility of constructing
"photonic crystals" with edge states that allow light to propagate in
one direction only, analogous to electronic quantum Hall edge states.   
      
Another important area of application of topological ideas has
been in the fractional quantum Hall effect, where they explain its
fractionally-charged excitations and their "fractional statistics";
FQHE states can be described as "topological field theories". The
current excitement in this field involves "non-Abelian" FQHE states,
where vortices in the electron fluid can fractionalize further into
intrinsically-entangled quasiparticles that it has been suggested can
be used to perform "topological quantum computing", as the
entanglement of widely-separated quasiparticles cannot be degraded by
coupling to the environment. The 5/2 FQHE state is widely believed to
be such a state, and current excitement centers on the attempt to
confirm this by interference experiments with non-Abelian quasiparticles.