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Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions

In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range...

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Autores principales: Seiringer, Robert, Yngvason, Jakob
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2020
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7462917/
https://www.ncbi.nlm.nih.gov/pubmed/32921810
http://dx.doi.org/10.1007/s10955-020-02586-0
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author Seiringer, Robert
Yngvason, Jakob
author_facet Seiringer, Robert
Yngvason, Jakob
author_sort Seiringer, Robert
collection PubMed
description In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range of the potential tends to zero while its strength tends to infinity. In a common approach the interaction potential is expanded in angular momentum eigenstates in the lowest Landau level, which amounts to taking the pre-factors to be the moments of the potential. Such a procedure is not appropriate for very strong interactions, however, in particular not in the case of hard spheres. We derive the formulas valid in the short-range case, which involve the scattering lengths of the interaction potential in different angular momentum channels rather than its moments. Our results hold for bosons and fermions alike and generalize previous results in [6], which apply to bosons in the lowest angular momentum channel. Our main theorem asserts the convergence in a norm-resolvent sense of the Hamiltonian on the whole Hilbert space, after appropriate energy scalings, to Hamiltonians with contact interactions in the lowest Landau level.
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spelling pubmed-74629172020-09-11 Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions Seiringer, Robert Yngvason, Jakob J Stat Phys Article In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range of the potential tends to zero while its strength tends to infinity. In a common approach the interaction potential is expanded in angular momentum eigenstates in the lowest Landau level, which amounts to taking the pre-factors to be the moments of the potential. Such a procedure is not appropriate for very strong interactions, however, in particular not in the case of hard spheres. We derive the formulas valid in the short-range case, which involve the scattering lengths of the interaction potential in different angular momentum channels rather than its moments. Our results hold for bosons and fermions alike and generalize previous results in [6], which apply to bosons in the lowest angular momentum channel. Our main theorem asserts the convergence in a norm-resolvent sense of the Hamiltonian on the whole Hilbert space, after appropriate energy scalings, to Hamiltonians with contact interactions in the lowest Landau level. Springer US 2020-06-24 2020 /pmc/articles/PMC7462917/ /pubmed/32921810 http://dx.doi.org/10.1007/s10955-020-02586-0 Text en © The Author(s) 2020 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Article
Seiringer, Robert
Yngvason, Jakob
Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions
title Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions
title_full Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions
title_fullStr Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions
title_full_unstemmed Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions
title_short Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions
title_sort emergence of haldane pseudo-potentials in systems with short-range interactions
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7462917/
https://www.ncbi.nlm.nih.gov/pubmed/32921810
http://dx.doi.org/10.1007/s10955-020-02586-0
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