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Open Systems & Information Dynamics

Composite Fermions in Braid Group Terms

Janusz Edward Jacak, Ireneusz Jóźwiak, Lucjan Jacak

Published Mar 1, 2010• 1 citation
Publisher

Abstract

A new implementation of composite fermions, and more generally — of composite anyons is formulated, exploiting one-dimensional unitary representations of appropriately constructed subgroups of the full braid group, in accordance with a cyclotron motion of 2D charged particle systems. The nature of hypothetical fluxes attached to the Jain's composite fermions is explained via additional cyclotron trajectory loops consistently with braid subgroup structure. It is demonstrated that composite fermions and composite anyons are rightful 2D particles (not an auxiliary construction) associated with cyclotron braid subgroups instead of the full braid group, which may open a new opportunity for non-Abelian composite anyons for quantum information processing applications.

DOI
10.1142/s1230161210000059

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A new implementation of composite fermions, and more generally — of composite anyons is formulated, exploiting one-dimensional unitary representations of appropriately constructed subgroups of the full braid group, in accordance with a cyclotron motion of 2D charged particle systems. The nature of hypothetical fluxes attached to the Jain's composite fermions is explained via additional cyclotron trajectory loops consistently with braid subgroup structure. It is demonstrated that composite fermions and composite anyons are rightful 2D particles (not an auxiliary construction) associated with cyclotron braid subgroups instead of the full braid group, which may open a new opportunity for non-Abelian composite anyons for quantum information processing applications.
It is shown that composite fermions—particles which appear in fractional quantum Hall effect, and are statistically different from anyons—could be described within braid group formalism. In the proposed model some ways of exchange of bare particles of composite particles are restricted. The new group describing exchanges of particles is defined. The description of such a group is included and the unitary one-dimensional representation of this group is found. An explicit form of this group for two exemplary cases is presented.
When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.
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 theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall theories, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy)1). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. The introduction into multi-component quantum Hall systems and spin quantum Hall systems is sketched. 1)First condensed states of composite fermion excitations have been very recently confirmed in the experiment (Pan et al. Phys. Rev. Lett. 90 (2003) 016801). a sample of this book is available at http://www.oup.co.uk/isbn/0-19-852870-1
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θθθ. θ=0,πθ= 0, πθ=0,π correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, θθθ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern--Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase θθθ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the ν=1/3ν= 1/3ν=1/3 fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
We present a careful construction of anyons on a torus starting with the braid-group analysis. The rules of Wen, Dagotto, and Fradkin for putting anyons on a torus are reproduced with some minor improvements. The existence of noncontractible loops leads to braid-group representations characterized not only by anyon statistics \ensuremath{\theta} but also by the magnetic fluxes ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ threading through the holes of the torus. The three parameters are tangled with each other. We explore the symmetries of the torus to separate the effects of ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ from those of \ensuremath{\theta}. It is shown that the anyon system always has a smaller period \ensuremath{\theta}/\ensuremath{\pi} in ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ than the natural period 1. We perform several numerical calculations to investigate the spectral flow and consistency of the method and find interesting features in the spectral flow, which are relevant in understanding the fractional quantum Hall effect.