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Gaps, Traps and Lattices - Correlations in Small Quantum Systems

Magnus O. Borgh

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

This dissertation investigates properties of two-dimensional many-body systems. Studies are performed using the Spin-Density Functional Theory with the Local Spin-Density Approximation, and numerical exact diagonalization. The properties studied include symmetry-breaking states in few-electron quantum dots, gaps in confined few-body systems, magnetic properties of cold fermionic atoms in optical lattices, and vortex formation in few-body systems. The dissertation also studies the properties of the SDFT-LSDA method for many-body calculations itself. The dissertation comprises five original papers, which are presented following an introduction to the fields of research, and the methods used, and the systems studied. Paper I investigates symmetry-breaking states in quantum dots, and the reliability of SDFT-LSDA for such systems. It is found that SDFT-LSDA may introduce artificial energy splittings between the members of a degenerate spin multiplet in the ground state. Paper II investigates the reliability of SDFT-LSDA when used to calculate gaps, and addition and removal energies in few-body systems. It is found that, contrary to studies of atoms and bilk solids, Kohn-Sham eigenvalues can be used to calculate addition and removal energies in parabolically confined systems. Also, in Paper II, van der Waals blockade is predicted to occur in systems of cold atoms, in analogy to Coulomb blockade in electronic systems. In Papers III and IV, mean-field theory is used to study the magnetic properties of cold fermionic atoms in an optical lattice. The results are compared with a similar study for quantum-dot lattices. It is found that there is a rich magnetic phase diagram, with non-magnetic, ferromagnetic, and anti-ferromagnetic states. The phase diagram of the optical lattice is found to be very similar to the phase diagram of the quantum-dot lattice. Paper V uses exact diagonalization to study vortex formation in rotating few-body quantum systems. Systems consisting of Coulomb-interacting fermions (electrons), Coulomb-interacting bosons, and bosons with a short-range interaction are studied. It is found that vortex formation in these systems has universal properties.

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This dissertation investigates properties of two-dimensional many-body systems. Studies are performed using the Spin-Density Functional Theory with the Local Spin-Density Approximation, and numerical exact diagonalization. The properties studied include symmetry-breaking states in few-electron quantum dots, gaps in confined few-body systems, magnetic properties of cold fermionic atoms in optical lattices, and vortex formation in few-body systems. The dissertation also studies the properties of the SDFT-LSDA method for many-body calculations itself. The dissertation comprises five original papers, which are presented following an introduction to the fields of research, and the methods used, and the systems studied. Paper I investigates symmetry-breaking states in quantum dots, and the reliability of SDFT-LSDA for such systems. It is found that SDFT-LSDA may introduce artificial energy splittings between the members of a degenerate spin multiplet in the ground state. Paper II investigates the reliability of SDFT-LSDA when used to calculate gaps, and addition and removal energies in few-body systems. It is found that, contrary to studies of atoms and bilk solids, Kohn-Sham eigenvalues can be used to calculate addition and removal energies in parabolically confined systems. Also, in Paper II, van der Waals blockade is predicted to occur in systems of cold atoms, in analogy to Coulomb blockade in electronic systems. In Papers III and IV, mean-field theory is used to study the magnetic properties of cold fermionic atoms in an optical lattice. The results are compared with a similar study for quantum-dot lattices. It is found that there is a rich magnetic phase diagram, with non-magnetic, ferromagnetic, and anti-ferromagnetic states. The phase diagram of the optical lattice is found to be very similar to the phase diagram of the quantum-dot lattice. Paper V uses exact diagonalization to study vortex formation in rotating few-body quantum systems. Systems consisting of Coulomb-interacting fermions (electrons), Coulomb-interacting bosons, and bosons with a short-range interaction are studied. It is found that vortex formation in these systems has universal properties.
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.
This dissertation, based on six included papers, theoretically investigates properties of quantum few-particle systems. An overview of related experimental research - ultra-cold trapped dilute gases, and electrons in quantum dots - is given, followed by a description of some of the studied many-particle phenomena - Bose-Einstein condensation, quantized vortices, Wigner localization and the Tonks-Girardeau gas. As the research results are presented in the included papers, a main part of the text in this thesis sets focus on methodology. Most of the papers involve use of the configuration interaction method, a numerical method which can give approximative eigenvalues and eigenstates of a few-particle Hamiltonian. The research has also involved further development of this method, by use of the Lee-Suzuki approximation. Formal descriptions of the methods are presented, together with a discussion about the numerical implementation. Explicit examples are given in an appendix. Papers I and II investigate properties of a rotating two-component Bose-Einstein condensate, in particular emerging vortex structures and associated wavefunctions. Paper III demonstrates that the Lee-Suzuki approximation, initially developed in the field of nuclear structure theory, can be useful to describe short-range particle-particle correlations in a trapped bosonic gas. Paper IV investigates the possibility to observe Wigner localization in a nanowire quantum dot, and compares predicted electron transport properties with experimental measurements. Paper V analyzes structures of ultra-cold atoms or molecules with dipolar interactions, in a quasi-one-dimensional trap. Paper VI also considers cold atoms or molecules with dipolar interactions, but in a quasi-two-dimensional setup, with a focus on the resulting Wigner states' dependence on the anisotropy of the interaction.
Quantum dot structures confine electrons in a small region of space. Some properties of semiconductor quantum dots, such as the discrete energy levels and shell filling effects visible in addition spectra, have analogies to those of atoms and indeed dots are sometimes referred to as 'artificial atoms'. However, atoms and dots show some fundamental differences due to electron correlations. For real atoms, the kinetic energy of electrons dominates over their mutual Coulomb repulsion energy and for this reason the independent electron approximation works well. For quantum dots the confining potential may be shallower than that of real atoms leading to lower electron densities and a dominance of mutual Coulomb repulsion over kinetic energy. In this strongly correlated regime the independent electron picture leads to qualitatively incorrect results. This thesis concentrates on few-electron quantum dots in the strongly correlated regime both for quasi-one-dimensional and two-dimensional dots in a square confining potential. In this so-called 'Wigner' regime the ground-state electronic charge density is localised near positions of classical electrostatic minima and the interacting electronic spectrum consists of well separated spin multiplets. In the strongly correlated regime the structure of low-energy multiplets is explained by mapping onto lattice models with extended-Hubbard and Heisenberg effective Hamiltonians. The parameters for these effective models are calculated within a Hartree approximation and are shown to reproduce well the exact results obtained by numerical diagonalisation of the full interacting Hamiltonian. Comparison is made between square dots and quantum rings with full rotational symmetry. In the very low-density regime, direct diagonalisation becomes impractical due to excessive computer time for convergence. In this regime a numerical renormalisation group method is applied to one-dimensional dots, enabling effective spin-interactions to be pr
This dissertation deals with finite-size effects in a few different quantum many-body phenomena in ultracold atomic systems. The finite-sized systems were simulated numerically using both mean-field methods and methods beyond mean-field, e.g. quadratic configuration interaction and exact diagonalization. The thesis is based on five scientific papers:Paper I is about the finite-size effects in the dynamics of a few-body soliton-like state. The collapses and revivals of the solitary wavefront in the particle density is characterized both analytically and numerically in the limit of weak interactions.Paper II analyzes a scheme to renormalize the contact interaction when used for exact diagonalization in a two-dimensional space. By relating the coupling strength in the diagonalization using the two-particle system, for which regularized solutions exists, converged results are obtained.Paper III deals with the formation of quantized vortices when stirring a harmonically trapped finite-sized system with a quadrupole deformation. I the energy spectrum the avoided crossing is found to be described by either of two types of correlated states. The type of avoided crossing can be controlled by the inclusion of an extra quartic deformation. Hysteresis in the mean-field description is found to be related to the types of avoided crossings exhibited and the full many-body time-evolution is compared to mean-field results. Paper IV investigates vortices in rotating fermionic droplets with dipole-dipole interactions. The vortex structure is found to still be present after the interaction is made anisotropic by having the dipoles tilt. Paper V contains an investigation of the effects on shell structure in fermionic droplets with dipole-dipole interactions. For a anisotropic harmonic oscillator, by tilting the dipoles, the shell structure of an isotropic oscillator can be restored.
We investigate universal properties of strongly confined particles that turn out to be dramatically different from what is observed for electrons in atoms and molecules. For a large class of harmonically confined systems, such as small quantum dots and optically trapped atoms, many-body particle addition and removal energies, and energy gaps, are accurately obtained from single-particle eigenvalues. Transport blockade phenomena are related to the derivative discontinuity of the exchange-correlation functional. This implies that they occur very generally, with Coulomb blockade being a particular realization of a more general phenomenon. In particular, we predict a van der Waals blockade in cold atom gases in traps.
In this thesis, quantum cluster methods are used to calculate electronic properties of correlated-electron systems. A special focus lies in the determination of the ground state properties of a 3/4 filled triangular lattice within the one-band Hubbard model. At this filling, the electronic density of states exhibits a so-called van Hove singularity and the Fermi surface becomes perfectly nested, causing an instability towards a variety of spin-density-wave (SDW) and superconducting states. While chiral d+id-wave superconductivity has been proposed as the ground state in the weak coupling limit, the situation towards strong interactions is unclear. Additionally, quantum cluster methods are used here to investigate the interplay of Coulomb interactions and symmetry-breaking mechanisms within the nematic phase of iron-pnictide superconductors. The transition from a tetragonal to an orthorhombic phase is accompanied by a significant change in electronic properties, while long-range magnetic order is not established yet. The driving force of this transition may not only be phonons but also magnetic or orbital fluctuations. The signatures of these scenarios are studied with quantum cluster methods to identify the most important effects. Here, cluster perturbation theory (CPT) and its variational extention, the variational cluster approach (VCA) are used to treat the respective systems on a level beyond mean-field theory. Short-range correlations are incorporated numerically exactly by exact diagonalization (ED). In the VCA, long-range interactions are included by variational optimization of a fictitious symmetry-breaking field based on a self-energy functional approach. Due to limitations of ED, cluster sizes are limited to a small number of degrees of freedom. For the 3/4 filled triangular lattice, the VCA is performed for different cluster symmetries. A strong symmetry dependence and finite-size effects make a comparison of the results from different clusters difficult. The ground state in the weak-coupling limit is superconducting with chiral d+id-wave symmetry, in accordance to previous renormalization group approaches. In the regime of strong interactions SDW states are preferred over superconductivity and a collinaer SDW state with nonuniform spin moments on a quadrupled unit cell has the lowest grand potential. At strong coupling, inclusion of short-range quantum fluctuations turns out to favor this collinear state over the chiral phase predicted by mean-field theory. At intermediate interactions, no robust conclusion can be drawn from the results. Symmetry-breaking mechanisms within the nematic phase of the iron-pnictides are studied using a three-band model for the iron planes on a 4-site cluster. CPT allows a local breaking of the symmetry within the cluster without imposing long-range magnetic order. This is a crucial step beyond mean-field approaches to the magnetically ordered state, where such a nematic phase cannot easily be investigated. Three mechanisms are included to break the fourfold lattice symmetry down to a twofold symmetry. The effects of anisotropic magnetic couplings are compared to an orbital ordering field and anisotropic hoppings. All three mechanisms lead to similar features in the spectral density. Since the anisotropy of the hopping parameters has to be very large to obtain similar results as observed in ARPES, a phonon-driven transition is unlikely.