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arXiv:cond-mat/0008286 · Journal of Physics A Mathematical and General

Exact solutions for interacting boson systems under rotation

Thomas Papenbrock, George F Bertsch

Published Jan 12, 2001• 13 citations
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Abstract

We study a class of interacting, harmonically trapped boson systems at angular momentum L . The Hamiltonian leaves an L -dimensional subspace invariant, and this permits an explicit solution of several eigenstates and energies for a wide class of two-body interactions.
DOI
10.1088/0305-4470/34/3/318

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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
A system of N two-dimensional weakly interacting bosons in a harmonic trap is considered. When the two-particle potential is a \ensuremath{\delta} function, Smith and Wilkin have proved analytically that the elementary symmetric polynomials of particle coordinates measured from the center of mass are exact eigenstates. In this study, we point out that their proof works equally well for an arbitrary two-particle potential that possesses translational and rotational symmetries. We find that the interaction energy associated with the eigenstate with angular momentum L is equal to $aN(N\ensuremath{-}1)/2+(b\ensuremath{-}a)NL/2,$ where a and b are the interaction energies of two bosons in the lowest-energy one-particle state with zero and one unit of angular momentum, respectively. Additionally, we study briefly the case of attractive quartic interactions. We prove rigorously that the lowest-energy state is the one in which all angular momentum is carried by the center of mass motion.
In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.
As a direct generalization of our previous work, we construct a manifestly rotation invariant Hamiltonian describing a system composed of a tensor boson with spin I = 2 — such as, for example, phonon of nuclear quadrupole surface vibration — in which special cubic and quartic self-interactions are taken into account. All the ground state solutions are obtained. It is shown that these solutions can be classified into three classes: The first is the normal vacuum of boson, which is manifestly rotation symmetry. The second class of solutions can be viewed as generated from a single chosen state by rotating it into an arbitrary direction. The third one is Goldstone modes built on this chosen ground state. This chosen state — an exact ground state solution of our rotation invariant Hamiltonian — is not an eigenstate of angular momentum, but contains infinitely many different angular momentum states in it; thus, typifying “spontaneous breakdown” of “rotation symmetry”. It is further shown that this state can be naturally identified to an “intrinsic” deformed state in nuclear collective model of Bohr and Mottelson. The results are examined from the standpoint of Goldstone theorem and detailed discussions are given to explore how Goldstone theorem is modified in our finite dimensional system.