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Physical Review B

Thermodynamic behavior of braiding statistics for certain fractional quantum Hall quasiparticles

Gun Sang Jeon, Jainendra K. Jain

Published Jan 12, 2010• 13 citations
Publisher

Abstract

It is widely believed that the braiding statistics of the quasiparticles of the fractional quantum Hall effect is a robust, topological property, independent of the details of the Hamiltonian or the wave function. However, for the quasiparticles of the 1/3 state, an explicit evaluation of the braiding phases using Laughlin's wave function has not produced a well-defined braiding statistics. We revisit this issue and demonstrate that the expected braiding statistics is recovered in the thermodynamic limit for exchange paths that are of finite extent but not for macroscopically large exchange loops that encircle a finite fraction of electrons. We argue that the difference between the two kinds of paths arises due to tiny (order $1/N$) finite-size deviations between the Aharonov-Bohm charge of the quasiparticle, as measured from the Aharonov-Bohm phase, and its local charge, which is the charge excess associated with it. An implication of our work is that models for quasiparticles that produce identical local charge can lead to different braiding statistics, which therefore can, in principle, be used to distinguish between such models.

DOI
10.1103/physrevb.81.035319

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It is widely believed that the braiding statistics of the quasiparticles of the fractional quantum Hall effect is a robust, topological property, independent of the details of the Hamiltonian or the wave function. However, for the quasiparticles of the 1/3 state, an explicit evaluation of the braiding phases using Laughlin's wave function has not produced a well-defined braiding statistics. We revisit this issue and demonstrate that the expected braiding statistics is recovered in the thermodynamic limit for exchange paths that are of finite extent but not for macroscopically large exchange loops that encircle a finite fraction of electrons. We argue that the difference between the two kinds of paths arises due to tiny (order 1/N1/N1/N) finite-size deviations between the Aharonov-Bohm charge of the quasiparticle, as measured from the Aharonov-Bohm phase, and its local charge, which is the charge excess associated with it. An implication of our work is that models for quasiparticles that produce identical local charge can lead to different braiding statistics, which therefore can, in principle, be used to distinguish between such models.
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We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic (mmn)(mmn)(mmn) FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations (FFF symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global Z2{\mathbb{Z}}_{2}Z2​ bilayer symmetry.
Quasihole excitations in fractional quantum Hall (FQH) systems exhibit fractional statistics and fractional spin, but how the spin-statistics relation emerges from many-body physics remains poorly understood. Here we prove a spin-statistics relation using only FQH wave functions, on both the sphere and disk geometry. In particular, the proof on the disk generalizes to all quasiholes in realistic systems, which have a finite size and could be deformed into arbitrary shapes. Different components of the quasihole spins are linked to different conformal Hilbert spaces (CHSs), which are null spaces of model Hamiltonians that host the respective FQH ground states and quasihole states. Understanding how the intrinsic spin of the quasiholes is linked to different CHSs is crucial for the generalized spin-statistics relation that takes into account the effect of metric deformation. In terms of the experimental relevance, this enables us to study the effect of deformation and disorder that introduces an additional source of Berry curvature, an aspect of anyon braiding that has been largely neglected in previous literature.