Cyclotron toroidal braids: A cure for portrayal composite fermions on a torus |
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| Institution: | 1. Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt;2. College of Business Administration in Majmaah, Majmaah University 11952, Kingdom of Saudi Arabia;1. Chaire Internationale de Physique Mathématique et Applications (CIPMA), Bénin;2. Département de Physique, Faculté des Sciences et Techniques (FAST), Bénin;3. Institut de Mathématiques et de Sciences Physiques (IMSP), Bénin;4. Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762–4900, USA;5. Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia;6. Department of Applied Mathematics, National Research Nuclear University, 31 Kashirskoe Hwy, Moscow 115409, Russian Federation;7. Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, Pretoria, South Africa;8. Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey;9. Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar;1. Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan;2. Department of Physics and Center for High Energy Physics, Chung-Yuan Christian University, Chung-Li 32023, Taiwan;3. Department of Physics, Pukyong National University (PKNU), Busan 608–737, Republic of Korea;1. Space Science Centre (ANGKASA), Institute of Climate Change (IPI), Universiti Kebangsaan Malaysia, 43600 UKM, Malaysia;2. Department of Electrical, Electronic & Systems Engineering, Faculty of Engineering & Built Environment, Universiti Kebangsaan Malaysia, Malaysia;3. Research Centre for Applied Physics and Radiation Technologies, Sunway University, Malaysia |
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| Abstract: | We present an inspection of the statistics of particles including composite fermions on a torus starting from a braid group analysis. For this purpose we considered a system of electrons confined to the surface of a torus under the influence of a strong magnetic field and interacting through a general rotational invariant potential. An explanation of the appearance of the cyclotron braids as an effect of restriction imposed by magnetic field on braid trajectories which in analyzed case reduces the full braid group to one of its subgroups (i.e. cyclotron subgroups), is given. The modified Feynman path-integral method is also reproduced with some minor enhancements. We improve known results concerning on braid groups on a torus in two directions: we obtain new estimates in terms of cyclotron braid subgroups and cyclotron band generator, respectively; we demonstrate that only multi-loop generators are accessible in the fractional quantum regime well and we also formally explain the unique statistic of composite fermions by construct trial wave function for the system on a torus, based on this idea. The topological oddness of torus geometry can be driven by shifting of electrons between the two different group of generators allowed for an explanation in satisfactory accordance the both compact commensurability condition and some numerical calculations in toroidal geometry. Besides, our approach may shed some new light on few interesting aspects in better understanding the fractional quantum Hall effect. |
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