In this paper, we recall the topological approach to quantum Hall effects. We note that, in the presence of a magnetic field, trajectories representing elements of the system’s braid group are of cyclotron orbit type. In two-dimensional spaces, this leads to the restriction of the full braid group, π1(Ω)—loopless generators (exchanges of MN coordinates or classical particles) are unenforceable. As a result, the identification of a possible Hall-like state comes down to the identification of a possible subgroup of π1(Ω). The latter follows from the connection between the one-dimensional unitary representation of the system’s braid group and particle statistics (unavoidable for any correlated state). In this work, we implement the topological approach to derive the lowest Landau-level pyramid of fillings. We point out that it contains all mysterious odd-denominator filling factors—like , or —not trivial to explain within the standard picture. We also introduce, explicitly, cyclotron subgroup generators for all derived fractions. Preliminary results on wave functions, supported by several Monte Carlo calculations, are presented. It is worth emphasizing that not all proposed many-body functions are purely antisymmetric—they, however, transform in agreement with the scalar representations of the system’s braid group. The latter is enforced by standard quantization methods.
- Received October 7, 2016.
- Accepted December 12, 2016.
- © 2017 The Author(s)
Published by the Royal Society. All rights reserved.