The quantum statistics of particles refers to the behavior
of a multiparticle wavefunction under adiabatic interchange
of two identical particles. While a three dimensional world
affords the possibilities of Bosons or Fermions, the two
dimensional world has more exotic possibilities such as
Fractional and Nonabelian statistics (J. Frölich, in ``Nonperturbative Quantum Field Theory", ed, G. t'Hooft. 1988). The latter is
perhaps the most interesting where the wavefunction obeys a
``nonabelian'' representation of the braid group -- meaning
that braiding A around B then B around C is not the same as
braiding B around C then A around B. This property enables
one to think about using these exotic systems for robust
topological quantum computation (M. Freedman, A. Kitaev, et al, Bull Am Math Soc 40, 31 (2003)). Surprisingly, it is thought that
quasiparticles excitations with such nonabelian
statistics may actually exist in certain quantum Hall states
that have already been observed. The most likely such
candidate is the quantum Hall \nu=5/2 state(R. L. Willett et al, Phys. Rev. Lett. 59, 1776-1779 (1987)), thought to be a
so-called Moore-Read Pfaffian state(G. Moore and N. Read, Nucl Phys. B360 362 (1991)), which can be thought of as a p-wave paired
superconducting state of composite fermions(M. Greiter, X. G. Wen, and F. Wilczek, PRL 66, 3205 (1991)). Using this superconducting
analogy, we use a Chern-Simons field theory approach to make
a number of predictions as to what experimental signatures
one should expect for this state if it really is this Moore-Read
state(K. Foster, N. Bonesteel, and S. H. Simon, PRL 91 046804 (2003)).
We will then discuss how the nonabelian statistics
can be explored in detail using a quantum monte-carlo
approach (Y. Tserkovnyak and S. H. Simon, PRL 90 106802 (2003)),
(I. Finkler, Y. Tserkovnyak, and S. H. Simon, work in progress.)
that allows one to explicitly drag one particle around another
and observe the change in the wavefunctions. Unfortunately,
it turns out that the Moore-Read state is not suited for
topological quantum computation\footnote[3]M. Freedman, A. Kitaev, et al, Bull Am Math Soc 40, 31 (2003). so we will turn our attention
to more the so-called parafermionic states(E. Rezayi and N. Read, Phys. Rev. B 59, 8084-8092 (1999).) which may also exist in nature.
[S3.002] Non Abelian States of Rotating Bose Gases
Nigel Cooper (University of Cambridge)
We study the groundstates of interacting Bose gases at high
angular momentum. We discuss both the case of
energy-independent repulsion, and the case where
interactions are strongly affected by a nearby Feshbach
resonance. We show that exact groundstates at high angular
momentum can be found analytically for a general and
realistic model for the resonant interactions. We identify
parameter regimes where the exact groundstates are exotic
fractional quantum Hall states, the excitations of which
obey non-abelian exchange statistics.
[S3.003] Conformal Field Theory Approaches to Quantum Hall States and Rotating Bose Condensates
Kareljan Schoutens (Institute for Theoretical Physics, University of Amsterdam)
Among quantum states of matter encountered in condensed
matter physics, fractional quantum Hall liquids are a
particularly intriguing category. Such states are
traditionally seen in 2D electron gases, where they are
responsible for the fractional quantum Hall effect. Novel
incarnations of quantum Hall liquids are expected for
rapidly rotating condensates of cold, bosonic or fermionic,
atoms. Numerical analysis has suggested that, for rapidly
rotating bosons, a particular series of non-abelian quantum
Hall states, first proposed by Read and Rezayi, is relevant.
These states are conveniently described and analyzed with
the help of Conformal Field Theory (CFT) techniques. We will
explain the CFT-quantum Hall connection, apply it to the
Read-Rezayi states, and present a variety of quantum Hall
spin liquids for spin-1/2 fermions and spin-1 bosons.
[S3.004] The Hanbury Brown-Twiss set-up for fractional statistics
Smitha Vishveshwara (University of Illinois at Urbana-Champaign)
The quasiparticle excitations of the fractional quantum Hall (FQH) system have been predicted to have fractional charge and statistics, and measurements for detecting fractional charge have been proposed and employed with success. Here, inspired by the Hanbury-Brown Twiss experiment, which revealed the bosonic statistics of photons some decades ago, a set-up is proposed for measuring fractional statistics in the FQH system. In this set-up it is shown, explicitly for Laughlin states, that edge-state quasiparticle current-current correlation measurements from two sources into two sinks can carry signatures of both fractional charge and fractional statistics. In addition, the consequences of such measurements for fractional states that cannot be described by Laughlin wavefunctions are discussed.