Symmetry breaking and quantum correlations in finite systems: studies of quantum dots and ultracold Bose gases and related nuclear and chemical methods

Abstract

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.

Nonuniversal Transmission Phase Lapses through a Quantum Dot: An Exact Diagonalization of the Many-Body Transport Problem

“…In summary, focusing on the mesoscopic regime of electron interferometry, and using the Bardeen weak-coupling theory in conjunction with exact diagonalization of the many-body quantum dot Hamiltonian, we have shown for the first two transitions, (a) N = 1 → N = 2 and (b) N = 2 → N = 3, nonuniversal behavior of the transmission phases with no phase lapse for (a) and a phase lapse of π for (b), in agreement with the experiment [2]. These results were obtained for a range of dot parameters characterized by shape anisotropy and strong e − e repulsion, with both favoring electron localization and formation of Wigner molecules [5,15,16,17]. Additionally, our analysis of the quasiparticle wavefunction [Eq.…”

Double-Well Ultracold-Fermions Computational Microscopy: Wave-Function Anatomy of Attractive-Pairing and Wigner-Molecule Entanglement and Natural Orbitals

“…In keeping with previous literature on two electrons in semiconductor quantum dots 21,30 , base 2 logarithms are used.…”

Symmetry-breaking in the independent particle model: nature of the singular behavior of Hartree–Fock potentials

“…We shall focus here on solutions preserving not only the total spin component S z but also the total spin S 2 , i.e., on RHF or ROHF solutions. The total-spin-violating solutions that arise in the presence of a triplet or a nonsinglet instability in closed-shell systems lead to UHF solutions that break the S 2 invariance and often also the space symmetry (as is, e.g., the case for quantum dots; see reviews [44,45]). Clearly, open-shell systems are always unstable to total-spin breaking in view of spin-polarization due to a different number of spin-up and spindown electrons.…”