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.

Classes of spontaneous symmetry breaking at zero and low magnetic fields in single quantum dots (QD's) and quantum dot molecules (QDM's) are discussed in relation to the ratio RW between the interelectron Coulomb repulsion and the harmonic confinement, using spin-and-Space unrestricted Hartree-Fock calculations. These include: Wigner crystallization for RW > 1, and formation of non-crystallized electron puddles localized on the individual dots in QDM's, as well as spin-density waves in single QD's, for RW < 1.Pacs Numbers: 73.20.Dx, 71.45.Lr Two-dimensional (2D) electron gases have provided (e.g., the fractional quantum Hall effect [1,2]), and continue to provide (e.g., a charge-density wave at higher Landau levels [3]) a source of discovery of remarkable many-body phenomena. Recently, 2D artificial quantum dots (QD's) and quantum dot molecules (QDM's) have become available, with the capability of controlling the dots' size, shape, and number N of electrons [4,5].Single QD's are commonly referred to as "artificial atoms", since interpretations of transport and capacitance experiments draw often on analogies between such artificial structures and natural atoms [4,5]. Underlying these analogies is an effective (circular) central mean field (CMF) picture, with the electronic spectra exhibiting (at zero magnetic field) shell closures and following Hund's rules for open shells. Indeed, in experiments on single QD's, the addition energy (AE) spectra [4] exhibit maxima at the expected closed shells (N = 2, 6, 12), and at the mid-shells (N = 4, 9, and 16) in accordance with Hund's rule.Here, using the self-consistent spin-and-Space unrestricted Hartree-Fock (sS-UHF) [6,7] method, we discuss, for zero and low magnetic fields (B), three types of spontaneous symmetry breakings (SB) in circular single QD's and in lateral QDM's (i.e., formation of ground states of lower symmetry than that of the confining potentials [12]). These include: (I) Wigner crystallization (WC) [13] in both QD's and QDM's, i.e., (spatial) localization of individual electrons, (II) formation of electron puddles (EP's) in QDM's, that is localization of the electrons on each of the individual dots comprising the QDM, but without crystallization within each dot, and (III) pure spin-density waves (SDW's) which are not accompanied by spatial localization of the electrons [9]). Furthermore, we show that CMF descriptions at zero and low magnetic fields may apply only for low values of the parameter R W ≡ Q/hω 0 , where Q is the Coulomb interaction strength andhω 0 is the parabolic confinement; Q = e 2 /κl 0 , with κ being the dielectric constant, l 0 = (h/m * ω 0 ) 1/2 the spatial extension of the lowest state's wave function in the parabolic confinement, and m * the effective electron mass. With the sS-UHF, we find that WC occurs (SB of type I) in both QD's and QDM's for R W > 1. For QDM's with R W < 1, WC does not develop and instead EP's may form (SB of type II). We note here that while certain quantum-mechanical studies of electron localization (WC a...

We present a group theoretical study of the symmetrybroken unrestricted Hartree-Fock orbitals and electron densities in the case of a two-dimensional N -electron single quantum dot (with and without an external magnetic field). The breaking of rotational symmetry results in canonical orbitals that (1) are associated with the eigenvectors of a Hückel hamiltonian having sites at the positions determined by the equilibrium molecular configuration of the classical N -electron problem, and (2) transform according to the irreducible representations of the point group specified by the discrete symmetries of this classical molecular configuration. Through restoration of the total-spin and rotational symmetries via post-Hartree-Fock projection techniques, we show that the point-group discrete symmetry of the unrestricted Hartree-Fock wave function underlies the appearance of magic angular momenta (familiar from exact-diagonalization studies) in the excitation spectra of the quantum dot. Furthermore, this two-step symmetrybreaking/symmetry-restoration method accurately describes the energy spectra associated with the magic angular momenta.

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A new class of analytic wave functions is derived for two dimensional N -electron (2 ≤ N < ∞) systems in high magnetic fields. These functions are constructed through breaking (at the Hartree-Fock level) and subsequent restoration (via post-Hartree-Fock methods) of the circular symmetry. They are suitable for describing long-range Coulomb correlations, while the Laughlin and composite-fermion functions describe Jastrow correlations associated with a short-range repulsion. Underlying our approach is a collectively-rotating-electronmolecule picture, yielding for all N an oscillatory radial electron density that extends throughout the system.

Investigations of the exactly solvable excitation spectra of two-electron quantum dots with a parabolic confinement, for different values of the parameter R(W) expressing the relative magnitudes of the interelectron repulsion and the zero-point kinetic energy, reveal for large R(W) a rovibrational spectrum associated with a linear trimeric rigid molecule composed of the two electrons and the infinitely heavy confining dot. This spectrum transforms to that of a "floppy" molecule for smaller R(W). The conditional probability distribution calculated for the exact two-electron wave functions allows identification of the rovibrational excitations as rotations and stretching/bending vibrations.

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The excitation spectrum of a two-electron quantum dot is investigated by tunneling spectroscopy in conjunction with theoretical calculations. The dot made from a material with negligible Zeeman splitting has a moderate spatial anisotropy leading to a splitting of the two lowest triplet states at zero magnetic field. In addition to the well-known triplet excitation at zero magnetic field, two additional excited states are found at finite magnetic field. The lower one is identified as the second excited singlet state on the basis of an avoided crossing with the first excited singlet state at finite fields. The measured spectra are in remarkable agreement with exact-diagonalization calculations. The results prove the significance of electron correlations and suggest the formation of a state with Wigner-molecular properties at low magnetic fields.