Exact solutions for interacting boson systems under rotation

Abstract

We study a class of interacting, harmonically trapped boson systems at angular momentum L. The Hamiltonian leaves a L-dimensional subspace invariant, and this permits an explicit solution of several eigenstates and energies for a wide class of two-body interactions. PACS numbers: 05.30.Jp, 03.75.Fi, 03.65.Fd, 67.40.Db Typeset using REVTEX 1 The study of vortices in Bose-Einstein condensates of trapped atomic vapors is of considerable experimental [1,2] and theoretical [3–13] interest. Present experiments and much theoretical work [3–7] focus on the Thomas-Fermi regime of short coherence length [14]. The opposite limit of perturbatively weak contact interactions is not yet realized experimentally but is of theoretical interest as well [8–13]. The case of attractive interactions was studied by Wilkin et al. [9]. Mottelson developed a theory for repulsive interactions [10], and Kavoulakis et al. [12] and Jackson et al. [13] compared mean field and exact numerical results. Exact diagonalization techniques by us showed that the ground state energy depends linearly on angular momentum and led to an analytical expression for the ground state wave function [11]. Recently, the form of this wave function and its eigenenergy were confirmed in analytic calculations [15–17]. However, we still lack insight about why certain eigenstates turn out to be simple analytical functions. We show in this paper that there is a subspace structure that explains these findings. Interestingly, this structure is not limited to the two-body contact interaction considered in previous investigations but present in a rather large class of Hamiltonians. Let us consider a system of N harmonically trapped bosons in two spatial dimensions 1 at angular momentum