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Physical Review B

Ground-state properties of quantum rings with a few electrons: Magnetization, persistent current, and spin chirality

Yuta Saiga, Dai S. Hirashima, J. Usukura

Published Jan 29, 2007• 31 citations
Publisher

Abstract

Ground-state properties of one-dimensional quantum rings with a few electrons, which interact with each other in the form of $1∕r$ Coulomb repulsion, are studied by exact diagonalization. For three electrons, the fully spin-polarized ground state is uniquely realized when the diameter of the ring is sufficiently large. In contrast, for four and five electrons, the fully polarized state never becomes the unique ground state, however large the diameter is. These results can be understood in terms of the multiple-spin exchange. When a perpendicular magnetic field is applied to the ring (i.e., the Aharonov-Bohm flux is introduced), the persistent current occurs, and the spin chirality is finite for a certain value of the flux. The difference between the quantum ring and the circularly arranged quantum dots is also discussed.

DOI
10.1103/physrevb.75.045343

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Ground-state properties of one-dimensional quantum rings with a few electrons, which interact with each other in the form of 1∕r1∕r1∕r Coulomb repulsion, are studied by exact diagonalization. For three electrons, the fully spin-polarized ground state is uniquely realized when the diameter of the ring is sufficiently large. In contrast, for four and five electrons, the fully polarized state never becomes the unique ground state, however large the diameter is. These results can be understood in terms of the multiple-spin exchange. When a perpendicular magnetic field is applied to the ring (i.e., the Aharonov-Bohm flux is introduced), the persistent current occurs, and the spin chirality is finite for a certain value of the flux. The difference between the quantum ring and the circularly arranged quantum dots is also discussed.
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.
We investigate the ground state persistent spin current and the pair entanglement in one-dimensional antiferromagnetic anisotropic Heisenberg ring with twisted boundary conditions. By solving Bethe ansatz equations numerically, we calculate the dependence of the ground state energy on the total magnetic flux through the ring, and the resulting persistent current. Motivated by the recent development of the quantum entanglement theory, we study the properties of the ground state concurrence under the influence of the flux through the anisotropic Heisenberg ring. We also include an external magnetic field and discuss the properties of the persistent current and the concurrence in the presence of the magnetic field.
The ground-state persistent current and electron addition spectrum in two-dimensional quantum dot arrays and one-dimensional quantum dot rings, pierced by an external magnetic flux, are investigated using the extended Hubbard model. The collective multidot problem is shown to map exactly into the strong-field noninteracting finite-size Hofstadter butterfly problem at the spin polarization transition. The finite-size Hofstadter problem is discussed, and an analytical solution for limiting values of flux is obtained. In weak fields we predict interesting flux periodic oscillations in the spin component along the quantization axis with a periodicity given by $\ensuremath{\nu}h/e$ $(\ensuremath{\nu}<~1)$. The sensitivity of the calculated persistent current to interaction and disorder is shown to reflect the intricacies of various Mott-Hubbard quantum phase transitions in two-dimensional systems: the persistent current is suppressed in the antiferromagnetic Mott-insulating phase governed by intradot Coulomb interactions; the persistent current is maximized at the spin density wave--charge density wave transition driven by the nearest-neighbor interdot interaction; the Mott-insulating phase persistent current is enhanced by the long-range interdot interactions to its noninteracting value; the strong suppression of the noninteracting current in the presence of random disorder is seen only at large disorder strengths; at half-filling even a relatively weak intradot Coulomb interaction enhances the disordered noninteracting system persistent current; in general, the suppression of the persistent current by disorder is less significant in the presence of the long-range interdot Coulomb interaction.
We study a finite-width two-dimensional quantum ring with a quadratic ddd-wave altermagnetic term, Tan--Inkson confinement, a perpendicular magnetic field, and independently tunable Aharonov--Bohm (AB) flux. The altermagnetic interaction mixes mmm and m±2m\pm2m±2 angular channels while preserving their parity. Circular symmetry renders the crystal orientation spectrally irrelevant and makes the two spin blocks exactly isospectral. The resulting same-parity anticrossings are resolved at low filling, and multichannel mixing is found near the Fermi level of highly occupied rings. Independent converged calculations for Ne=1400N_e=1400Ne​=1400 and 180018001800 establish the reconstructed Fermi edge and magnetic response in a finite field window and the persistent current over a full AB flux period. The analytic conventional reference separates radial-subband depopulation (dHvA-type cycles) from finer finite-ring shell oscillations; we do not assign a global Fourier spectrum to unverified altermagnetic field scans. The few-electron thermal response is obtained from an exact fixed-particle-number canonical partition function, avoiding finite-system errors of a fixed-mean-number approximation. The circular model supports an equilibrium charge current but no persistent spin current.
We consider a one-dimensional mesoscopic quantum ring filled with spinless electrons and threaded by a magnetic flux, which carries a persistent current at zero temperature. The interplay of Coulomb interactions and a single on-site impurity yields a nontrivial dependence of the persistent current on the size of the ring. We determine numerically the asymptotic power law for systems up to 32 00032\text{ }00032 000 sites for various impurity strengths and compare with predictions from Bethe ansatz solutions combined with bosonization. The numerical results are obtained using an improved functional renormalization-group (fRG) method. We apply the density-matrix renormalization-group (DMRG) and exact diagonalization methods to benchmark the fRG calculations. We use DMRG to study the persistent current at low electron concentrations in order to extend the validity of our results to quasicontinuous systems. We briefly comment on the quality of calculated fRG ground-state energies by comparison with exact DMRG data.
We apply exact-exchange spin-density functional theory in the Krieger-Li-Iafrate approximation to interacting electrons in quantum rings of different widths. The rings are threaded by a magnetic flux that induces a persistent current. A weak space and spin symmetry breaking potential is introduced to allow for localized solutions. As the electron-electron interaction strength described by the dimensionless parameter r(S) is increased, we observe-at a fixed spin magnetic moment-the subsequent transition of both spin sub-systems from the Fermi liquid to the Wigner crystal state. A dramatic signature of Wigner crystallization is that the persistent current drops sharply with increasing r(S). We observe simultaneously the emergence of pronounced oscillations in the spin-resolved densities and in the electron localization functions indicating a spatial electron localization showing ferrimagnetic order after both spin sub-systems have undergone the Wigner crystallization. The critical r(S)(c) at the transition point is substantially smaller than in a fully spin-polarized system and decreases further with decreasing ring width. Relaxing the constraint of a fixed spin magnetic moment, we find that on increasing r(S) the stable phase changes from an unpolarized Fermi liquid to an antiferromagnetic Wigner crystal and finally to a fully polarized Fermi liquid.