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arXiv:1112.2197 · Journal of Statistical Mechanics Theory and Experiment

Exact results for the spectra of interacting bosons and fermions on the lowest Landau level

Stefan V. Mashkevich, Sergey I. Matveenko, Stéphane Ouvry

Published Feb 5, 2013• 2 citations
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Abstract

A system of N interacting bosons or fermions in a two-dimensional harmonic potential (or, equivalently, magnetic field) whose states are projected onto the lowest Landau level is considered. Generic expressions are derived for matrix elements of any interaction, in the basis of angular momentum eigenstates. For the fermion "ground state" (N=1 Laughlin state), this makes it possible to exactly calculate its energy all the way up to the mesoscopic regime N ~ 1000. It is also shown that for N = 3 and Coulomb interaction, several rational low-lying values of energy exist, for bosons and fermions alike.

DOI
10.1088/1742-5468/2013/02/p02013

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A system of N interacting bosons or fermions in a two-dimensional harmonic potential (or, equivalently, magnetic field) whose states are projected onto the lowest Landau level is considered. Generic expressions are derived for matrix elements of any interaction, in the basis of angular momentum eigenstates. For the fermion "ground state" (N=1 Laughlin state), this makes it possible to exactly calculate its energy all the way up to the mesoscopic regime N ~ 1000. It is also shown that for N = 3 and Coulomb interaction, several rational low-lying values of energy exist, for bosons and fermions alike.
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We study the rotational properties of a two-component Bose gas trapped in a rotating harmonic potential, effectively in two dimensions. We first consider a completely homogeneous interaction between particles, and assume the bosons to be in the lowest Landau level. We derive analytical expressions for some wave functions and energies at low angular momenta, and use these results together with exact numerical diagonalization results to test the applicability of composite fermion (CF) trial wave functions to the system. We find that a general procedure based on CF wave functions reproduces the exact ground states and many of the excited states for sufficiently low angular momenta, and give very good approximate results at angular momenta up to the two-component analogy of a single vortex. Additionally, we produce very simple CF states that have remarkably high overlaps with the lowest-lying states. Finally, we briefly discuss the case of inhomogeneous interactions, pointing out the difficulties arising from this generalization.