General Hartree-Fock method and symmetry breaking in quantum dots Fabio Cavaliere a,1 , Umberto De Giovannini b a LAMIA CNR-INFM, Dipartimento di Fisica, Universit`a di Genova, Via Dodecaneso 33, 16146 Genova, Italy b School of Engineering and Sciences, Jacobs University Bremen, Campus Ring 1, 28759 Bremen Germany Abstract Interaction and correlation effects in quantum dots play a fundamental role in defining both their equilibrium and transport properties. Numerical methods are commonly employed to study such systems. In this paper we present a two-step approach in which a Hartree–Fock method, with explicit symmetry breaking, is followed by a projection technique for symmetry restoration. Three different Hartree-Fock implementations, with an increasing degree of symmetry breaking, are introduced and applied to the study of interacting planar dots with N = 3 and N = 6, electrons in the presence of a perpendicular magnetic field. In addition to the restricted and unrestricted techniques already employed for quantum dots, the general unrestricted Hartree-Fock method is described. It is characterized by a complete breaking of all spatial and spin symmetries and improved energy estimates of the ground state energy. Projection techniques suitable for all three Hartree-Fock methods are introduced, and shown to generate correlated many-body wavefunctions. Key words: Quantum dots, Spin, Hartree Fock, Symmetry breaking, Symmetry restoration, Correlations PACS: 73.23.Hk, 73.63.Kv 1. Introduction In quantum dots [1], the interplay between spin and Coulomb interactions [2] markedly affects the trans- port properties, leading to such phenomena as spin blockade [3], peculiar conductance modulations [4–8], and negative differential conductance [9,10]. Strong electron–electron interactions may also lead to the formation of highly correlated molecular states of electrons [11–16], the finite–size analogues of Wigner crystals [17]. The study of interaction effects in quantum dots is a very important and remarkably tough problem [2], to which many numerical tools have been applied. The 1 Corresponding author. E-mail: cavalier@fisica.unige.it most accurate ones are exact diagonalizations [18– 20], recently employed also for the study of transport properties of dots with N ≤ 3 electrons [21], and quantum Monte Carlo [11–13,22] methods. They both produce correlated wavefunctions (WF) and allow to obtain energy spectra and quantum numbers. Due to their computational complexity, however, they are usually limited to small particle numbers. More manageable methods such as density functional the- ory [23,24] and Hartree–Fock (HF) [25–29] allow to explore higher electron numbers. Recently, the non– equilibrium Green’s function technique has also been proposed [30]. All these methods, though, are less accurate and exhibit breaking of the spatial and spin symmetries [15,23,28,30], which prevents the determi- nation of the quantum numbers of dot states. Preprint submitted to Physica E 24 April 2009