Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions
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Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions Robert Seiringer1
· Jakob Yngvason2
Received: 31 January 2020 / Accepted: 4 June 2020 © The Author(s) 2020
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.
1 Introduction In a seminal paper [1] on the fractional quantum Hall effect1 F.D.M. Haldane introduced two-body interaction operators (pseudo-potentials) that have Laughlin’s wave functions [5] as exact eigenstates. In suitable units the latter are the functions
1 A standard reference on the quantum Hall effect is [2]; see also the reviews [3] and [4].
Communicated by Alessandro Giuliani. 2020 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes
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Robert Seiringer [email protected] Jakob Yngvason [email protected]
1
IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
2
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
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R. Seiringer , J. Yngvason
mL (x1 , . . . , x N ) = C N ,m
(z i − z j )m
i< j
e−|xi |
2 /2
(1.1)
i
(1)
(2)
(1)
(2)
where N ≥ 2 is the particle number, the z j = x j + ix j ∈ C with x j = (x j , x j ) ∈ R2 are complex coordinates for the particles moving in a two-dimensional plane perpendicular to a strong magnetic field, m is a positive integer (even for bosons, odd for fermions), and C N ,m a normalization factor. The bosonic case, considered in [6] in the lowest angular momentum channel, is in particular relevant for dilute quantum gases in rapid rotation, where the rotational velocity takes the role of the magnetic field [7,8]. Haldane’s pseudo-potential operators have the form () Pi j (1.2) i= j ()
where Pi j , for a nonnegative integer , is the projector onto states in the lowest Landau level (LLL) with relative angular momentum for a pair i j. Recall th
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