Eur. Phys. J. D 16, 381–385 (2001) T HE EUROPEAN P HYSICAL JOURNAL D c  EDP Sciences Societ` a Italiana di Fisica Springer-Verlag 2001 Magnetic properties of quantum dots and rings M. Manninen 1, a , M. Koskinen 1 , S.M. Reimann 2 , and B. Mottelson 3 1 Department of Physics, University of Jyv¨askyl¨a, P.O. Box 35, FIN-40351 Jyv¨askyl¨a, Finland 2 Department of Mathematical Physics, Lund University, P.O. Box 118, SE-22100 Lund, Sweden 3 NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received 5 December 2000 Abstract. Exact many-body methods as well as current-spin-density functional theory are used to study the magnetism and electron localization in two-dimensional quantum dots and quasi-one-dimensional quantum rings. Predictions of broken-symmetry solutions within the density functional model are confirmed by exact configuration interaction (CI) calculations: In a quantum ring the electrons localize to form an antiferromagnetic chain which can be described with a simple model Hamiltonian. In a quantum dot the magnetic field localizes the electrons as predicted with the density functional approach. PACS. 73.21.La Quantum dots – 73.43.Nq Quantum phase transitions – 85.35.Be Quantum well devices (quantum dots, quantum wires, etc.) 1 Introduction The electronic structure of quantum dots has been an ex- tensive area of research during the last decades [1]. The simple harmonic confinement modeling a quantum dot makes the system especially well suited for applying shell model techniques: The center of mass motion exactly sep- arates out and, on the other hand, a harmonic oscillator basis is a natural starting point [2] for many-body calcu- lations. When the local spin density approximation has been used to study Hund’s rule and the magnetic structure of quantum dots [3], it has been observed that broken sym- metry solutions can result. Static spin-density waves of the ground states [3] and the localization caused by a strong magnetic field [4] have, however, been disputed [5,6] as being artifacts of mean field theory since the circular sym- metry of the exact Hamiltonian was broken. In this paper we will first study the spin-density wave in a six electron quantum ring and show that the re- sult of an exact many-body calculation can be mapped to a model Hamiltonian consisting of an antiferromag- netic Heisenberg Hamiltonian, combined with rigid rota- tions and vibrations. This result confirms the existence of the spin-density wave observed earlier using the local spin density approximation [3,8]. We then study Hund’s rule and electron localization of a four electron quantum dot as a function of the electron density. Finally, we will com- pare the results of exact calculations for six electron dots in magnetic field with those obtained with the current- spin-density formalism, as it originally was developed by a e-mail: manninen@phys.jyu.fi Vignale and Rasolt [7]. For a non-circular dot, we are able to observe the localization caused by a strong magnetic field also in the exact electron density. 2 The model The electrons are restricted to move in a plane and con- fined by an external potential (in atomic units) V (r)= 1 2 ω 0 (r - r 0 ) 2 , (1) where ω 0 is the strength of the confinement and r 0 the radius of the quantum ring (r 0 = 0 for a quantum dot). The electrons interact with each other with the normal 1/r Coulomb interaction. The many-body Hamiltonian is diagonalized using a configuration-interaction (CI) tech- nique. The spatial single-particle states of the Fock space are chosen to be eigenstates of the single particle part of the Hamiltonian. We expand them in the harmonic os- cillator basis. From 30 to about 50 lowest energy single- particle states are selected to span the Fock space. To set up the Fock states for diagonalization, we sample over the full space with a fixed number of spin down and spin up electrons. Only those states with a given total orbital an- gular momentum and configuration energy (corresponding to the sum of occupied single-particle energies) less than a specific cut-off energy are selected. The purpose was to choose only the most important Fock states from the full basis and hereby reducing the matrix dimension to a size < 2 × 10 5 . In the case of a deformed quantum dot in a magnetic field the restriction of the orbital angular mo- mentum could of course not be used.