https://doi.org/10.1140/epjp/s13360-022-03023-9
Regular Article
Spherical Artin groups in 2D quantum Hall system framework: a duality-based approach to description particle statistics in a sphere geometry
Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland
Received:
21
June
2021
Accepted:
30
June
2022
Published online:
19
July
2022
We analyze the braid trajectories existing on the surface of a sphere as compact locally 2D manifold. A radial magnetic field imposes restriction on braid trajectories, leading to the appearance cyclotron subgroups instead of the full braid group. Anyons statistic has been encoded in the well-known spherical Artin groups and extended to cyclotron braid subgroups and cyclotron band generator, as derived elements of the full braid group. We have also focused on isomorphism between the plane and the sphere. The modified Feynman path integral method is the background to the relevance of defined (cyclotron) braid trajectories, providing a quantum mechanical description, even though a classical trajectories are investigated. To demonstrate a magnetic field commensurability condition for a sphere, we use more mathematical and less illustrative way embedding it in one of the corollaries described in the paper. The standard quantization methods lead to the determination of the many-body wave functions for anyons population that transform in agreement with the scalar representations of the system’s braid group. The composite fermions statistic is also investigated within the presented idea and portrayed by new proposed ansatz.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2022
