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arXiv:cond-mat/0605321 · Journal of Physics Condensed Matter

Spectral properties of rotating electrons in quantum dots and their relation to quantum Hall liquids

M. Koskinen, S. M. Reimann, J.-P. Nikkarila, M. Manninen

Published Feb 2, 2007• 13 citations
arXivHTMLPDF

Abstract

The exact diagonalization technique is used to study many-particle properties of interacting electrons with spin, confined in a two-dimensional harmonic potential. The single-particle basis is limited to the lowest Landau level. The results are analysed as a function of the total angular momentum of the system. Only at angular momenta corresponding to the filling factors 1, 1/3, 1/5, etc is the system fully polarized. The lowest energy states exhibit spin waves, domains, and localization, depending on the angular momentum. Vortices exist only at excited polarized states. The high angular momentum limit shows localization of electrons and separation of the charge and spin excitations.

DOI
10.1088/0953-8984/19/7/076211

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The exact diagonalization technique is used to study many-particle properties of interacting electrons with spin, confined in a two-dimensional harmonic potential. The single-particle basis is limited to the lowest Landau level. The results are analysed as a function of the total angular momentum of the system. Only at angular momenta corresponding to the filling factors 1, 1/3, 1/5, etc is the system fully polarized. The lowest energy states exhibit spin waves, domains, and localization, depending on the angular momentum. Vortices exist only at excited polarized states. The high angular momentum limit shows localization of electrons and separation of the charge and spin excitations.
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.
In this dissertation, we investigate the edge spin excitations and reconstructions of quantum Hall (QH) liquids. Edge spin reconstructions reflect the interplay of confining potential, electron-electron interaction, and Zeeman splitting at the edges of QH liquids. The main method used in this dissertation is the exact diagonalization calculation for finite size systems. The QH liquids studied in this dissertation are classified into three kinds, based on their topological properties and statistics of quasiparticles (quasiholes). The first kind of QH liquids is spin-polarized QH liquids. The QH liquids with filling factors ν = 1 and 1/3 are studied in this kind. We find the low-energy excitations of the ν = 1 and 1/3 are bosonic edge spin waves. Instabilities of these ferromagnetic state with altering confinement strength result from the softening of these spin waves and formation of edge spin textures. A microscopic trial wave function is used to study the edge instabilities of larger systems with particle number up to 40, and the obtained results are consistent to the ones calculated by exact diagonalization. The second kind of QH liquids is spin-unpolarized QH liquids. The liquids with ν = 2, 2/5 and 2/3 are studied in this kind. When ν = 2, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial ν = 2 instability is toward the compact state with total spin 1. At ν = 2/5, the edge instabilities is also triggered by softening of the edge spin waves with smoother confinement. The situation of ν = 2/3 unpolarized QH liquids is a little different, because it has one backward-moving edge spin wave mode and one forward-moving magnetoplasmon mode. There is no mystery about what kind of instability would occur with stronger or smoother confinement. The exact diagonalization calculation of finite systems can reproduce the instability induced by edge spin wave at the stronger confinement side and gives an estimate about the critical value of confining potential. But instability due to edge magnetoplasmon mode can not be realized in small size systems because of the high velocity of this mode. The last kind is the non-Abelian QH liquid (at ν = 5/2). Its edge instability triggered by softening of edge magnetoplasmon excitations with smoother confinement. Three different confinements are compared in the study of integer QH liquids in finite size systems. And some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit. The effect of Zeeman splitting to the edge instabilities is also considered in this dissertation.
We study the effect of electron-electron interaction on the charge and spin structures at the edge of integer quantum Hall liquids, under three different kinds of confining potentials. Our exact diagonalization calculation for small systems indicates that the low-energy excitations of the $\ensuremath{\nu}=1$ ferromagnetic state are bosonic edge spin waves. Instabilities of the ferromagnetic state with altering confinement strength result from the softening of these edge spin waves and formation of edge spin textures. In the $\ensuremath{\nu}\ensuremath{\lesssim}2$ regime, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial $\ensuremath{\nu}=2$ instability is toward the compact state with total spin 1. Larger systems are studied using a microscopic trial wave function, and some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit.
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.
The magnetic properties of a two-dimensional interacting electron system in a strong magnetic field are considered in the quantum Hall regime, where the motion of the electrons is restricted to the lowest Landau level. We apply an approach based on the many-particle quantum-field theory which we developed recently for the spin-polarized system in the fractional quantum Hall regime, and which treats the problem of the degeneracy of a partially filled Landau level properly. Via a gauge transformation of the fermion Green's function, we achieve a dressing of the propagator with multiple collective excitations. Our theory allows a calculation of the magnetic polarization M(T)M(T)M(T) for higher temperatures TTT, where $M(T)\ensuremath{\lesssim}0.4$ is not too large. For filling factor $\ensuremath{\nu}=1$ we obtain a magnetic polarization M(T)M(T)M(T) which agrees quite well with the data of a NMR experiment by Barrett [Phys. Rev. Lett. 74, 5112 (1995)].
We present a theoretical treatment of four two-dimensional electrons in a harmonic confinement potential in the presence of an external magnetic field using the exact diagonalization approach. The ground state properties and the spin and angular momentum transitions for different electron interaction strengths and magnetic fields are obtained. A magnetic field-confinement strength phase diagram is presented indicating a rich variety of ground states. An interesting feature of this system is the depolarization of spins by application of a magnetic field. The results are compared to several approximate theories.