This chapter describes how, in two-dimensional systems, the scalar fractional statistics may occur besides Bose–Einstein and Fermi–Dirac statistics. One-dimensional irreducible unitary representations of the full braid group for selected manifolds (Euclidean plane, sphere, torus, and Euclidean three-dimensional space) and its relation with symmetry of the wave function of the system are presented. The idea of anyons — particles obeying fractional statistics — is discussed in detail. The relation between spin and statistics in two- and three-dimensional systems and its topological nature is discussed. The chapter also shows how magnetic flux attached to particles in planar systems can change their statistical properties.This chapter describes how, in two-dimensional systems, the scalar fractional statistics may occur besides Bose–Einstein and Fermi–Dirac statistics. One-dimensional irreducible unitary representations of the full braid group for selected manifolds (Euclidean plane, sphere, torus, and Euclidean three-dimensional space) and its relation with symmetry of the wave function of the system are presented. The idea of anyons — particles obeying fractional statistics — is discussed in detail. The relation between spin and statistics in two- and three-dimensional systems and its topological nature is discussed. The chapter also shows how magnetic flux attached to particles in planar systems can change their statistical properties.

Oxford University Press eBooksSep 25, 20030 citations

Fractional Statistics in Quantum Mechanics

Daniel P. Arovas

Series of the Centro de Estudios Científicos de SantiagoJan 1, 19920 citations

Statistical phases and momentum spacings for one-dimensional anyons

Martin Greiter

Anyons and fractional statistics are by now well established in two-dimensional systems. In one dimension, fractional statistics has been established so far only through Haldane's fractional exclusion principle, but not via a fractional phase the wave function acquires as particles are interchanged. At first sight, the topology of the configuration space appears to preclude such phases in one dimension. Here we argue that the crossings of one-dimensional anyons are always unidirectional, which makes it possible to assign phases consistently and hence to introduce a statistical angle $\ensuremath{\theta}$. The fractional statistics then manifests itself in fractional spacings of the single-particle momenta of the anyons when periodic boundary conditions are imposed. These spacings are given by $\ensuremath{\Delta}p=2\ensuremath{\pi}\ensuremath{\hbar}/L(|\ensuremath{\theta}|/\ensuremath{\pi}+\text{non-negative}\text{ }\text{integer})$ for a system of length $L$ . This condition is the analog of the quantization of relative angular momenta according to ${l}_{\text{z}}=\ensuremath{\hbar}(\ensuremath{-}\ensuremath{\theta}/\ensuremath{\pi}+2\ifmmode\times\else\texttimes\fi{}\text{integer})$ for two-dimensional anyons.Anyons and fractional statistics are by now well established in two-dimensional systems. In one dimension, fractional statistics has been established so far only through Haldane's fractional exclusion principle, but not via a fractional phase the wave function acquires as particles are interchanged. At first sight, the topology of the configuration space appears to preclude such phases in one dimension. Here we argue that the crossings of one-dimensional anyons are always unidirectional, which makes it possible to assign phases consistently and hence to introduce a statistical angle $\ensuremath{\theta}$. The fractional statistics then manifests itself in fractional spacings of the single-particle momenta of the anyons when periodic boundary conditions are imposed. These spacings are given by $\ensuremath{\Delta}p=2\ensuremath{\pi}\ensuremath{\hbar}/L(|\ensuremath{\theta}|/\ensuremath{\pi}+\text{non-negative}\text{ }\text{integer})$ for a system of length $L$. This condition is the analog of the quantization of relative angular momenta according to ${l}_{\text{z}}=\ensuremath{\hbar}(\ensuremath{-}\ensuremath{\theta}/\ensuremath{\pi}+2\ifmmode\times\else\texttimes\fi{}\text{integer})$ for two-dimensional anyons.

arXiv:0707.1011Physical Review BFeb 11, 200943 citations

Fractional Statistics and Quantum Theory

Avinash Khare

WORLD SCIENTIFIC eBooksJan 1, 2005124 citations

Fractional Statistics and Quantum Theory

Avinash Khare

WORLD SCIENTIFIC eBooksFeb 1, 2005113 citations

An ansatz for the exclusion statistics parameters in macroscopic physical systems described by fractional exclusion statistics

Dragoş-Victor Anghel

I introduce an ansatz for the exclusion statistics parameters of fractional-exclusion-statistics (FES) systems and I apply it to calculate the statistical distribution of particles from both, bosonic and fermionic perspectives. To check the applicability of the ansatz, I calculate the FES parameters in three well-known models: a Fermi-liquid–type system, a one-dimensional quantum system described in the thermodynamic Bethe ansatz and quasi-particle excitations in a fractional quantum Hall (FQH) system. The FES parameters of the first two models satisfy the ansatz, whereas those of the third model, although close to the form given by the ansatz, represent an exception. In this case I also show that the general properties of the FES parameters, deduced elsewhere ( EPL , 87 (2009) 60009), are satisfied also by the parameters of the FQH liquid.I introduce an ansatz for the exclusion statistics parameters of fractional-exclusion-statistics (FES) systems and I apply it to calculate the statistical distribution of particles from both, bosonic and fermionic perspectives. To check the applicability of the ansatz, I calculate the FES parameters in three well-known models: a Fermi-liquid–type system, a one-dimensional quantum system described in the thermodynamic Bethe ansatz and quasi-particle excitations in a fractional quantum Hall (FQH) system. The FES parameters of the first two models satisfy the ansatz, whereas those of the third model, although close to the form given by the ansatz, represent an exception. In this case I also show that the general properties of the FES parameters, deduced elsewhere ( EPL , 87 (2009) 60009), are satisfied also by the parameters of the FQH liquid.

arXiv:0909.0030Europhysics Letters (EPL)Apr 1, 201011 citations

L

. This condition is the analog of the quantization of relative angular momenta according to ${l}_{\text{z}}=\ensuremath{\hbar}(\ensuremath{-}\ensuremath{\theta}/\ensuremath{\pi}+2\ifmmode\times\else\texttimes\fi{}\text{integer})$ for two-dimensional anyons.