Seiringer, RobertISTA ; Yngvason, Jakob
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 , 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.
Journal of Statistical Physics
Open access funding provided by Institute of Science and Technology (IST Austria). The work of R.S. was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 694227). J.Y. gratefully acknowledges hospitality at the LPMMC Grenoble and valuable discussions with Alessandro Olgiati and Nicolas Rougerie.
Seiringer R, Yngvason J. Emergence of Haldane pseudo-potentials in systems with short-range interactions. Journal of Statistical Physics. 2020;181:448-464. doi:10.1007/s10955-020-02586-0
Seiringer, R., & Yngvason, J. (2020). Emergence of Haldane pseudo-potentials in systems with short-range interactions. Journal of Statistical Physics. Springer. https://doi.org/10.1007/s10955-020-02586-0
Seiringer, Robert, and Jakob Yngvason. “Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions.” Journal of Statistical Physics. Springer, 2020. https://doi.org/10.1007/s10955-020-02586-0.
R. Seiringer and J. Yngvason, “Emergence of Haldane pseudo-potentials in systems with short-range interactions,” Journal of Statistical Physics, vol. 181. Springer, pp. 448–464, 2020.
Seiringer R, Yngvason J. 2020. Emergence of Haldane pseudo-potentials in systems with short-range interactions. Journal of Statistical Physics. 181, 448–464.
Seiringer, Robert, and Jakob Yngvason. “Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions.” Journal of Statistical Physics, vol. 181, Springer, 2020, pp. 448–64, doi:10.1007/s10955-020-02586-0.
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