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arXiv:cond-mat/0410598 · Physical Review Letters

Crystalline Boson Phases in Harmonic Traps: Beyond the Gross-Pitaevskii Mean Field

Igor Romanovsky, Constantine Yannouleas, Uzi Landman

Published Dec 1, 2004• 51 citations
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Abstract

Strongly-interacting bosons in two-dimensional harmonic traps are described through breaking of rotational symmetry at the Hartree-Fock level and subsequent symmetry restoration via projection techniques, thus incorporating correlations beyond the Gross-Pitaevskii (GP) solution. The bosons localize and form polygonal-ringlike crystalline patterns, both for a repulsive contact potential and a Coulomb interaction, as revealed via conditional-probability-distribution analysis. For neutral bosons, the total energy of the crystalline phase saturates in contrast to the GP solution, and its spatial extent becomes smaller than that of the GP condensate. For charged bosons, the total energy and dimensions approach the values of classical pointlike charges in their equilibrium configuration.

DOI
10.1103/physrevlett.93.230405

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Strongly-interacting bosons in two-dimensional harmonic traps are described through breaking of rotational symmetry at the Hartree-Fock level and subsequent symmetry restoration via projection techniques, thus incorporating correlations beyond the Gross-Pitaevskii (GP) solution. The bosons localize and form polygonal-ringlike crystalline patterns, both for a repulsive contact potential and a Coulomb interaction, as revealed via conditional-probability-distribution analysis. For neutral bosons, the total energy of the crystalline phase saturates in contrast to the GP solution, and its spatial extent becomes smaller than that of the GP condensate. For charged bosons, the total energy and dimensions approach the values of classical pointlike charges in their equilibrium configuration.
We discuss the formation of crystalline electron clusters in semiconductor quantum dots and of crystalline patterns of neutral bosons in harmonic traps. In a first example, we use calculations for two electrons in an elliptic quantum dot to show that the electrons can localize and form a molecular dimer. The calculated singlet-triplet splitting (J) as a function of the magnetic field (B) agrees with cotunneling measurements with its behavior reflecting the effective dissociation of the dimer for large B. Knowledge of the dot shape and of J(B) allows determination of the degree of entanglement. In a second example, we study strongly repelling neutral bosons in two-dimensional harmonic traps. Going beyond the Gross-Pitaevskii (GP) mean-field approximation, we show that bosons can localize and form polygonal-ring-like crystalline patterns. The total energy of the crystalline phase saturates in contrast to the GP solution, and its spatial extent becomes smaller than that of the GP condensate.
We exploit rotational-symmetry breaking in the one-body density to examine the formation of structures in systems of N strongly coupled charged bosons with logarithmic repulsions inside isotropic two-dimensional harmonic traps, with N in the range from 2 to 7. The results serve as a map for ordered arrangements of vortices in a trapped Bose-Einstein condensate. Two types of N-body wavefunctions are assumed: (i) a permanent vertical bar psi(WM)> of N identical Gaussian orbitals centred at variationally determined sites, and (ii) a permanent vertical bar psi(SM)> of N orthogonal orbitals built from harmonic-oscillator energy eigenstates. With increasing coupling strength, the bosons in the vertical bar psi(WM)> orbitals localize into polygonal-ringlike crystalline patterns ('Wigner molecules'), whereas the wavefunctions vertical bar psi(SM)> describe low energy excited states containing delocalized bosons as in supersolid crystallites ('supermolecules'). For N = 2 at strong coupling both states describe a Wigner dimer.
We present a variational many-body wave function for repelling bosons in rotating traps, focusing on rotational frequencies that do not lead to restriction to the lowest Landau level. This wave function incorporates correlations beyond the Gross-Pitaevskii (GP) mean-field approximation, and it describes rotating boson molecules (RBMs) made of localized bosons that form polygonal-ring-like crystalline patterns in their intrinsic frame of reference. The RBMs exhibit characteristic periodic dependencies of the ground-state angular momenta on the number of bosons in the polygonal rings. For small numbers of neutral bosons, the RBM ground-state energies are found to be always lower than those of the corresponding GP solutions, in particular, in the regime of GP vortex formation.
The Hamiltonian for a small number $N\ensuremath{\leqslant}11$ of bosons in a rapidly rotating harmonic trap, interacting via a short range (contact potential) or a long range (Coulomb) interaction, is studied via an exact diagonalization in the lowest Landau level. Our analysis shows that, for both low and high fractional fillings, the bosons localize and form rotating boson molecules (RBMs) consisting of concentric polygonal rings. Focusing on systems with the number of trapped atoms sufficiently large to form multiring bosonic molecules, we find that, as a function of the rotational frequency and regardless of the type of repulsive interaction, the ground-state angular momenta grow in specific steps that coincide with the number of localized bosons on each concentric ring. Comparison of the conditional probability distributions (CPDs) for both interactions suggests that the degree of crystalline correlations appears to depend more on the fractional filling $\ensuremath{\nu}$ than on the range of the interaction. The RBMs behave as nonrigid rotors, i.e., the concentric rings rotate independently of each other. At filling fractions $\ensuremath{\nu}<1∕2$, we observe well developed crystallinity in the CPDs (two-point correlation functions). For larger filling fractions $\ensuremath{\nu}>1∕2$, observation of similar molecular patterns requires consideration of even higher-order correlation functions.
The mean-field properties of trapped, two-dimensional interacting Bose gases of finite size are surveyed numerically. The coupled nonlinear equations of the Gross-Pitaevskii (GP) equation and the Hartree-Fock equation for the condensate and the thermal component, respectively, are calculated by the hybrid method, in which the GP equation and the low-lying excitations less than cut-off energy are calculated quantum mechanically using the basis representation, while the high-lying states are obtained within the local density approximation. Consistent solutions are obtained for the whole range of temperatures including the critical temperature by determining the thermodynamic chemical potential µ from the condition that the total atomic number must be the desired value, instead of identifying µ with the eigenvalue of the GP equation. Comparisons with the results of the semiclassical two-fluid model are presented.
Investigations of emergent symmetry breaking phenomena occurring in small finite-size systems are reviewed, with a focus on the strongly correlated regime of electrons in two-dimensional semiconductor quantum dots and trapped ultracold bosonic atoms in harmonic traps. Throughout the review we emphasize universal aspects and similarities of symmetry breaking found in these systems, as well as in more traditional fields like nuclear physics and quantum chemistry, which are characterized by very different interparticle forces. A unified description of strongly correlated phenomena in finite systems of repelling particles (whether fermions or bosons) is presented through the development of a two-step method of symmetry breaking at the unrestricted Hartree-Fock level and of subsequent symmetry restoration via post Hartree-Fock projection techniques. Quantitative and qualitative aspects of the two-step method are treated and validated by exact diagonalization calculations.Strongly-correlated phenomena emerging from symmetry breaking include the following.Chemical bonding, dissociation and entanglement (at zero and finite magnetic fields) in quantum dot molecules and in pinned electron molecular dimers formed within a single anisotropic quantum dot, with potential technological applications to solid-state quantum-computing devices.Electron crystallization, with particle localization on the vertices of concentric polygonal rings, and formation of rotating electron molecules (REMs) in circular quantum dots. Such electron molecules exhibit ro-vibrational excitation spectra, in analogy with natural molecules.At high magnetic fields, the REMs are described by parameter-free analytic wave functions, which are an alternative to the Laughlin and composite-fermion approaches, offering a new point of view of the fractional quantum Hall regime in quantum dots (with possible implications for the thermodynamic limit).Crystalline phases of strongly repelling bosons. In rotating traps and in analogy with the REMs, such repelling bosons form rotating boson molecules (RBMs). For a small number of bosons, the RBMs are energetically favored compared with the Gross-Pitaevskii solutions describing vortex formation.We discuss the present status concerning experimental signatures of such strongly correlated states, in view of the promising outlook created by the latest experimental improvements that are achieving unprecedented control over the range and strength of interparticle interactions.