Cyclotron braid group structure for composite fermions

Abstract

Although they describe properties of 2D Hall systems in the fractional quantum regime well, composite fermions suffer from the unexplained character of the localized magnetic field flux-tubes attached to each particle in order to reproduce the Laughlin correlations via Aharonov-Bohm phase shifts. The identification of the cyclotron trajectories of 2D charged particles as accessible classical trajectories within the braid group approach at the magnetic field presence, allows, however, for the avoidance of the construction with fluxes. We introduce cyclotron braid subgroups for charged 2D systems at the fractional Landau-level filling associated in a more natural way with composite fermions without invoking field flux-tubes. The Aharonov-Bohm phase shifts caused by fluxes are replaced with the phase gain due to multi-loop cyclotron trajectories unavoidably occurring at the fractional filling of 1/p (p is an odd integer). Another approach to composite particles, using so-called vortices, is also discussed from the point of view of the cyclotron braid group description (for both odd and even p integers).

Topological approach to quantum Hall effects and its important applications: higher Landau levels, graphene and its bilayer

“…The topological approach to quantum Hall effects -called the cyclotron subgroup model [16][17][18][20][21][22][23] -is entirely based on the mathematical concept of braid groups. The full braid group of a multi-particle system contains all closed trajectories encircled in an appropriate configuration space and organised in homotopy classes [24,25],…”

“…For example, one can obtain bosons, σ i = e i0 , but also fermions, σ i = e iπ , using 1DURs of the permutation group S N . If the full braid group has a far more rich structure -like in the case of 2D manifolds -the analyzation of possible 1DURs may lead to an infinite number of quantum particles (including bosons and fermions) with fractional statistics, e iθ with θ ∈ [0, 2π) [16,17,[20][21][22]. The latter are called anyons.…”

“…Note that some external factors may change the form of braid group describing the system, by -for instancepreventing the realization of some types of (classes of homotopical) trajectories [16][17][18][19][20][21][22][23]. This effect (the reshaping of a group) occurs in two-dimensional manifolds after the inclusion of a strong magnetic field -when a classical cyclotron motion confines the variety of accessible braids.…”

“…Furthermore, a cyclotron trajectory is always a closed path and its simplest version is obtained with the use of two multi-loop exchanges (when a statistics is determined and a system is described with a certain subgroup). Finally, the cyclotron trajectory (as a mapping of a braid group element from Ω on a M manifold) has p loops -this quantity is equal to the total number of external magnetic field flux quanta attributed to a particle in the case of The one-dimensional unitary representations of a cyclotron subgroup have the same form as 1DURs of π 1 restricted to this subgroup [16,17,[20][21][22],…”

“…What is more, the presence -in bilayer systems [13][14][15] -of incompressible ν = 1 2 was also confusing for Jain's theory supporters. For all of these reasons, in this paper, we introduce the cyclotron subgroups model [16][17][18][19][20][21][22][23], which is based on the topology of twodimensional systems upon strong magnetic fields, to the issue of the FQHE in higher Landau levels, in monolayer and bilayer graphene.…”