Eur. Phys. J. B (2017) 90: 90
https://doi.org/10.1140/epjb/e2017-80065-0

Regular Article

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

Faculty of Fundamental Problems of Technology, Wroclaw University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

^a e-mail: patrycja.lydzba@pwr.edu.pl

Received: 25 January 2017
Received in final form: 16 March 2017
Published online: 18 May 2017

Abstract

In this paper the topological approach to quantum Hall effects is carefully described. Commensurability conditions together with proposed generators of a system braid group are employed to establish the fractional quantum Hall effect hierarchies of conventional semiconductors, monolayer and bilayer graphene structures. Obtained filling factors are compared with experimental data and a very good agreement is achieved. Preliminary constructions of ground-state wave functions in the lowest Landau level are put forward. Furthermore, this work explains why pyramids of fillings from higher bands are not counterparts of the well-known composite-fermion hierarchy – it provides with the cause for an intriguing robustness of ν = 7/3 , 8/3 and 5/2 states (also in graphene). The argumentation why paired states can be developed in two-subband systems (wide quantum wells) only when the Fermi energy lies in the first Landau level is specified. Finally, the paper also clarifies how an additional surface in bilayer systems contributes to an observation of the fractional quantum Hall effect near half-filling, ν = 1/2 .