New implementation of composite fermions in terms of subgroups of a braid group

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.…”

“…In the standard, Jain's [4,5], model of the fractional quantum Hall effect it is impossible to distinguish composite from ordinary fermions, because they both experience the same quantum statistics [16,20]. This is due to the fact that 1DURs are periodic with a periodicity of 2π and so e ipπ = e iπ .…”

“…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.…”

“…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.…”