Electron and boson clusters in confined geometries: Symmetry breaking in quantum dots and harmonic traps

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   Fig. 1.

   UHF electron density in a parabolic QD for N = 19 electrons and S _z = 19/2, exhibiting breaking of the circular symmetry at R _W = 5 and zero magnetic field. Remaining parameters: parabolic confinement, ℏω₀ = 5 meV; effective mass, m* = 0.067m_e. Distances are in nanometers, and the electron density is in 10⁻⁴ nm⁻².

   
   
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   Fig. 2.

   The ST splitting J = E^s − E^t as a function of the magnetic field B for an elliptic QD with ℏω_x = 1.2 meV and ℏω_y = 3.3 meV (these values correspond to the device of ref. 22). Solid line, GHL (broken-symmetry UHF plus restoration of symmetries) results with a coordinate-independent screening (κ = 22); dashed line, EXD results with κ = 12.9 (GaAs) but including screening with a coordinate dependence according to ref. 24 and d = 18.0 nm (see text). Remaining material parameters: m*(GaAs) = 0.067m _e, and g* = 0 (see text). The experimental measurements (22) are denoted by open squares. Our sign convention for J is opposite of that in ref. 22.

   
   
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   Fig. 3.

   Total electron densities (EDs) associated with the singlet state of the elliptic dot at B = 0 and B = 2.5 T. (a) The GHL densities. (b) The EXD densities. The rest of the parameters and the screening of the Coulomb interaction are as in Fig. 2. Lengths are in nanometers, and densities are in 10⁻⁴ nm⁻².

   
   
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   Fig. 4.

   Von Neumann entropy for the singlet state of the elliptic dot as a function of the magnetic field B. Solid line, GHL; dashed line, EXD. The rest of the parameters and the screening of the Coulomb interaction are as in Fig. 2. At the top, we show histograms for the |z_k|² coefficients (see Eq. 8) of the singlet state at B = 1.3 T, illustrating the dominance of two configurations. Note the small third coefficient |z ₃|² = 0.023 in the EXD case.

   
   
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   Fig. 5.

   Total energies as a function of R _δ for various approximation levels, calculated for N = 6 harmonically confined 2D bosons in the (1,5) lowest-energy configuration. RBHF/G, restricted Bose–Hartree–Fock (RBHF) energy, E _RBHF ^G _, with the common orbital φ₀(r) approximated by a Gaussian centered at the trap origin; GP, the Gross–Pitaevskii energy; PRJ, the energy of the symmetry-restored state obtained via projection of the (unrestricted) UBHF state. Energies are in units of ℏω₀.

   
   
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   Fig. 6.

   Single-particle densities (a–c) and conditional probability distribution (d) for N = 6 2D harmonically trapped neutral bosons with a contact interaction and R _δ = 25. (a) The single-orbital self-consistent GP case. (b) The symmetry-broken UBHF case (static crystallite). (c) The projected case (symmetry-restored wave function; see Eq. 11). The crystalline structure of the outer ring in this last case is “hidden” but is revealed in the conditional probability distribution (10, 16) displayed in d, where the observation point is denoted by a black dot (on the right). Lengths are in units of l ₀.