Homotopy braid group description including cyclotron motion of charged interacting two-dimensional (2D) particles at strong magnetic field presence is developed in order to explain, in algebraic topology terms, Laughlin correlations in fractional quantum Hall systems. There are introduced special cyclotron braid subgroups of a full braid group with 1D unitary representations suitable to satisfy Laughlin correlation requirements. In this way an implementation of composite fermions (fermions with auxiliary flux quanta attached in order to reproduce Laughlin correlations) is formulated within uniform for all 2D particles braid group approach. The fictitious fluxes — vortices attached to the composite fermions in a traditional formulation are replaced with additional cyclotron trajectory loops unavoidably occurring when ordinary cyclotron radius is too short in comparison to particle separation and does not allow for particle interchanges along single-loop cyclotron braids. Additional loops enhance the effective cyclotron radius and restore particle interchanges. A new type of 2D particles — composite anyons is also defined via unitary representations of cyclotron braid subgroups. It is demonstrated that composite fermions and composite anyons are rightful 2D particles, not auxiliary compositions with fictitious fluxes and are associated with cyclotron braid subgroups instead of the full braid group, which may open also a new opportunity for non-Abelian composite anyons for topological quantum information processing applications, due to richer representations of subgroup than of a group.
"Cyclotron braid group approach to Laughlin correlations." Adv. Theor. Math. Phys. 15 (2) 449 - 469, April 2011.