Abstract
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.
Highlights
In a seminal paper [1] on the fractional quantum Hall effect1 F.D.M
We introduce the sequence of closed subspaces
For fixed γ and λ small enough the unique ground state equals the Laughlin state L+1, where h has eigenvalue 0. It is separated from other states with the same or less angular momentum by a spectral gap since the state space is finite dimensional, but the size of the gap might a-priori depend on the particle number N
Summary
In a seminal paper [1] on the fractional quantum Hall effect F.D.M. Haldane introduced two-body interaction operators (pseudo-potentials) that have Laughlin’s wave functions [5] as exact eigenstates. Wave functions with finer structure, avoiding configurations where a bona fide interaction potential is very large, must necessarily have components in higher Landau levels These components may tend to zero as the range tends to zero, the joint effects of the kinetic and potential energies at length scales much smaller than the magnetic length will leave a trace in the LLL, and lead in particular to a replacement of v as the coupling constant in front of (1.2) by a constant proportional to the s-wave scattering length of v, as rigorously established in [6]. Our main theorem states that after suitable -dependent energy scaling the full Hamiltonian converges in this sense to a Hamiltonian in the lowest Landau level with the pseudo-potential interaction operator (1.2) and a definite coupling constant. No rigorous justification for the pseudo-potential description in the Coulomb case is known,
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