We explore the correlations and entanglements of exact-diagonalized few-electron wave functions in a quantum dot in magnetic fields without the Zeeman splitting. With the increase of the field, the lowest states with different spins gradually form a narrow band and the electronic states undergo a transition from liquids to rotating Wigner molecules which are accompanied by different characters of charge correlations. For both the liquid and crystal states, the spin conditional probability densities show magnetic couplings between the particles which depend on the particle numbers, the total spins and the angular momenta of the states. The von Neumann entropies show the spin-dependent entanglements between electrons. The regular magnetic-coupling oscillations and converging entanglement entropies emerge in the rotating Wigner molecular states.
We explore the correlations and entanglements of exact-diagonalized few-electron wave functions in a quantum dot in magnetic fields without the Zeeman splitting. With the increase of the field, the lowest states with different spins gradually form a narrow band and the electronic states undergo a transition from liquids to rotating Wigner molecules which are accompanied by different characters of charge correlations. For both the liquid and crystal states, the spin conditional probability densities show magnetic couplings between the particles which depend on the particle numbers, the total spins and the angular momenta of the states. The von Neumann entropies show the spin-dependent entanglements between electrons. The regular magnetic-coupling oscillations and converging entanglement entropies emerge in the rotating Wigner molecular states.
Exact-diagonalization calculations for N=3 electrons in anisotropic quantum dots, covering a broad range of confinement anisotropies and strength of interelectron repulsion, are presented for zero and low magnetic fields. The excitation spectra are analyzed as a function of the strength of the magnetic field and for increasing quantum-dot anisotropy. Analysis of the intrinsic structure of the many-body wave functions through spin-resolved two-point correlations reveals that electrons tend to localize forming Wigner molecules. For certain ranges of dot parameters (mainly at strong anisotropy), the Wigner molecules acquire a linear geometry, and the associated wave functions with a spin projection Sz=1∕2 are similar to the representative class of strongly entangled states referred to as the W states. For other ranges of parameters (mainly at intermediate anisotropy), the Wigner molecules exhibit a more complex structure consisting of two mirror isosceles triangles. This latter structure can be viewed as an embryonic unit of a zigzag Wigner crystal in quantum wires. The degree of entanglement in three-electron quantum dots can be quantified through the use of the von Neumann entropy.
Energy spectra, spin configurations, and entanglement characteristics of a system of four electrons in lateral double quantum dots are investigated using exact diagonalization (EXD), as a function of interdot separation, applied magnetic field, and strength of interelectron repulsion. A distinctly different quantum behavior is found compared to that of circular single quantum dots. As a function of the magnetic field, the energy spectra exhibit a low-energy band consisting of a group of six states, with the number six being a consequence of the conservation of the total spin of the four electrons and the ensuing spin degeneracies. These six states appear to cross at a single value of the magnetic field, with the crossing point becoming sharper for larger interdot distances. As the strength of the Coulomb repulsion increases, the six states tend to become degenerate and a well defined energy gap separates them from the higher-in-energy excited states. The appearance of the low-energy band is a consequence of the formation of a Wigner supermolecule, with the four electrons (two in each dot) being localized at the vertices of a rectangular parallelogram. Using the spin-resolved pair-correlation functions, one can map the EXD many-body wave functions onto the spin functions associated with the four localized electrons. Analogies with nanomagnets, such as finite Heisenberg clusters, are discussed. The ability to determine associated spin functions enables investigations concerning entanglement properties of the system of four electrons. In particular, the formation of Wigner supermolecules generates EXD solutions belonging to the class of strongly entangled states referred to as N-qubit Dicke states in the quantum-information literature.
The trial wavefunctions for few-electron quantum rings are presented to describe the spin-dependent rotating Wigner molecule states. The wavefunctions are constructed from the single-particle orbits which contain two variational parameters to describe the shape and size dependence of electron localization in the ring-like confinement. They can explicitly show the size dependence of single-particle orbital occupation to give an understanding of the spin rules of ground states without magnetic fields. They can also correctly describe the spin and angular momentum transitions in magnetic fields. By examining the von Neumann entropy, it is demonstrated that the wavefunctions can illustrate the entanglement between electrons in quantum rings, including the AB oscillations as well as the spin and size dependence of the entropy. Such trial wavefunctions will be useful in investigating spin-related quantum behaviors of a few electrons in quantum rings.
The studies on quantum dots and rings containing $4\ensuremath{\sim}6$ electrons reveal that the shapes of the systems have great impact on the spin transitions of the ground states. For five-electron case, the different shapes lead to the different transition processes in quantum dots and rings. For six-electron case, the ring confinement forbids the spin transition, which really exists in dot confinement. The investigations on the angular momentum transitions in magnetic fields show that the sensibility of the transitions to the size of the system depends remarkably on the particle number. The four-electron case is almost size insensitive but the five-electron case has obvious size dependence. It is also found that the behaviors of the few-electron entanglement in quantum dots and rings are quite different. The entanglement entropies in quantum dots increase with the increase in the magnetic field but exhibit Aharonov-Bohm oscillation patterns in narrow rings. The studies of the shape, size, and particle number effects are important for understanding the liquid-to-crystal transitions, and may be helpful for the manipulation of electronic states in quantum dots and rings.
Energy spectra and spin configurations of a system of N=4 electrons in lateral double quantum dots (quantum dot helium molecules) are investigated using exact diagonalization (EXD), as a function of interdot separation, applied magnetic field (B), and strength of interelectron repulsion. As a function of the magnetic field, the energy spectra exhibit a low-energy band consisting of a group of six states, with the number six being a consequence of the conservation of the total spin and the ensuing spin degeneracies for four electrons. The energies of the six states appear to cross at a single value of the magnetic field, and with increasing Coulomb repulsion they tend to become degenerate, with a well-defined energy gap separating them from the higher-in-energy excited states. The appearance of the low-energy band is a consequence of the formation of a Wigner supermolecule, with the four electrons (two in each dot) being localized at the vertices of a rectangle. Using spin-resolved pair-correlation distributions, a method for mapping the complicated EXD many-body wave functions onto simpler spin functions associated with a system of four localized spins is introduced. Detailed interpretation of the EXD spin functions and EXD spectra associated with the low-energy band via a four-site Heisenberg cluster (with B-dependent exchange integrals) is demonstrated. Aspects of spin entanglement, referring to the well-known N-qubit Dicke states, are also discussed.
This thesis investigates the use of wave-function methods for the study of quantum-dot systems. It investigates single dots, using quantum Monte Carlo techniques in a wide range of magnetic field values, and a double-dot system, using the exact-diagonalization method.\n\nThe thesis proposes simple yet accurate many-particle wave functions for various angular-momentum and spin states, for both weak and strong magnetic fields. Using these trial wave functions, it evaluates various properties of dots and studies Wigner crystallization and spin polarization for the weak-field limit. For strong magnetic fields, the thesis investigates ground states of different spin polarizations as a function of the magnetic field and tests the commonly used lowest-Landau-level approximation. The results are compared to calculations from the density-functional theory.\n\nFinally, the thesis presents a method that combines the accuracy of the exact-diagonalization method and the scalability of Monte Carlo methods. The Monte Carlo-based diagonalization is a promising tool for use in situations that cannot be handled with a simple trial wave function, and have too many particles for an exact-diagonalization treatment. For quantum dots, methods for efficiently evaluating the high-magnetic-field basis functions and their gradients are demonstrated.