We present a brief introduction of the fractional quantum Hall effect—a description of the phenomenon is provided and basic requirements for its formation are discussed. We recall assumptions of the standard composite fermion theory. Additionally, we present a list of the fractional quantum Hall effect puzzles. The chapter also introduces the non-local approach to quantum Hall physics, which is entirely based on a mathematical concept of braid groups and their reduction stimulated by an external magnetic field (in two-dimensional spaces). We emphasize the connection between a one-dimensional unitary representation of the system braid group and the particle statistics (unavoidable for any correlated Hall-like states). We implement our topological approach to construct hierarchies of FQHE fillings for various two-dimensional structures, including multi-layers. We show the remarkable agreement of our results with experimental findings.
Part of the book: Recent Advances in Graphene Research