An Anyon Primer

The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The main formalism used in many-body quantum Hall theories -- the Chern-Simons theory is also presented. The Chern-Simons the...

Anyons, particles that are neither bosons nor fermions, were predicted in the 1980s, but strong experimental evidence for the existence of the simplest type on anyons has only emerged this year. Further theoretical and experimental advances promise to nail the existence of more exotic types on anyons, such as Majorana fermions, which would make topological quantum computation possible.

Quantum mechanical systems, whose degrees of freedom are so-called su(2)_k anyons, form a bridge between ordinary SU(2) spin systems and systems of interacting non-Abelian anyons. Such a connection can be made for arbitrary spin-S systems, and we explicitly discuss spin-1/2 and spin-1 systems. Anyonic spin-1/2 chains exhibit a topological protection mechanism that stabilizes their gapless ground states and which vanishes only in the limit (k to infinity) of the ordinary spin-...

Strongly correlated fractional quantum Hall liquids support fractional excitations, which can be understood in terms of adiabatic flux insertion arguments. A second route to fractionalization is through the coupling of weakly interacting electrons to topologically nontrivial backgrounds such as in polyacetylene. Here we demonstrate that electronic fractionalization combining features of both these mechanisms occurs in noncoplanar itinerant magnetic systems, where integer quan...

In this paper, we report on the study of Abelian and non-Abelian statistics through Fabry-Perot interferometry of fractional quantum Hall (FQH) systems. Our detection of phase slips in quantum interference experiments demonstrates a powerful, new way of detecting braiding of anyons. We confirm the Abelian anyonic braiding statistics in the $\nu = 7/3$ FQH state through detection of the predicted statistical phase angle of $2\pi/3$, consistent with a change of the anyonic part...

Studying quantum entanglement in systems of indistinguishable particles, in particular anyons, poses subtle challenges. Here, we investigate a model of one-dimensional anyons defined by a generalized algebra. This algebra has the special property that fermions in this model are composites of anyons. A Hubbad-like Hamiltonian is considered that allows hopping between nearest neighbour sites not just for the fundamental anyons, but for the fermionic anyon composites. Some inter...

We study the Quantum Field Theory of nonrelativistic bosons coupled to a Chern--Simons gauge field at nonzero particle density. This field theory is relevant to the study of anyon superconductors in which the anyons are described as {\bf bosons} with a statistical interaction. We show that it is possible to find a mean field solution to the equations of motion for this system which has some of the features of bose condensation. The mean field solution consists of a lattice of...

We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (`identity') channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional conformal f...

The fractional quantum Hall effect is a paradigm of topological order and has been studied thoroughly in two dimensions. Here, we construct a new type of fractional quantum Hall system, which has the special property that it lives in fractal dimensions. We provide analytical wave functions and exact few-body parent Hamiltonians, and we show numerically for several different Hausdorff dimensions between 1 and 2 that the systems host anyons. We also find examples of fractional ...

According to a basic rule of fermionic and bosonic many-body physics, known as the linked cluster theorem, physical observables are not affected by vacuum bubbles, which represent virtual particles created from vacuum and self-annihilating without interacting with real particles. Here, we show that this conventional knowledge must be revised for anyons, quasiparticles that obey fractional exchange statistics intermediate between fermions and bosons. We find that a certain cla...