2D magnetic stability

This article is a contribution to the proceedings of the 33rd/35th International Colloquium on Group Theoretical Methods in Physics (ICGTMP, Group33/35) held in Cotonou, Benin, July 15-19, 2024. The stability of matter is an old and mathematically difficult problem, relying both on the uncertainty principle of quantum mechanics and on the exclusion principle of quantum statistics. We consider here the stability of the self-interacting almost-bosonic anyon gas, generalizing the Gross-Pitaevskii / nonlinear Schrödinger energy functionals to include magnetic self interactions. We show that there is a type of supersymmetry in the model which holds only for higher values of the magnetic coupling but is broken for lower values, and that in the former case supersymmetric ground states exist precisely at even-integer quantized values of the coupling. These states constitute a manifold of explicit solitonic vortex solutions whose densities solve a generalized Liouville equation, and can be regarded as nonlinear generalizations of Landau levels. The reported work is joint with Alireza Ataei and Dinh-Thi Nguyen and makes an earlier analysis of self-dual abelian Chern-Simons-Higgs theory by Jackiw and Pi, Hagen, and others, mathematically rigorous.

💡 Research Summary

The paper investigates the stability of a two‑dimensional gas of almost‑bosonic anyons when the particles generate and interact with their own magnetic field. Starting from the many‑body Hamiltonian that incorporates Aharonov‑Bohm fluxes attached to each particle, the author reviews the classical results on stability of matter—namely the roles of the Heisenberg uncertainty principle, the Pauli exclusion principle, and Lieb‑Thirring type inequalities. By adapting recent Lieb‑Thirring bounds for anyons, the author proves that the system enjoys “stability of the second kind” (the ground‑state energy grows at most linearly with particle number) provided the statistical parameter (\alpha) does not belong to the even‑integer lattice.

The core of the work is a mean‑field functional that extends the Gross‑Pitaevskii (GP) energy to include a self‑generated magnetic vector potential. For a one‑body wave function (u) with density (\rho=|u|^{2}), the magnetic potential is defined by a Biot–Savart convolution

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