Journal of Magnetism and Magnetic Materials 272–276 (2004) e687–e688 Induced anyonicity and consequences A. Ralko*, T.T. Truong Laboratoire de Physique Th ! eorique et Mod " elisation, Universit ! e de Cergy-Pontoise (CNRS/UMR 8089), F-95031 Cergy-Pontoise Cedex, France Abstract We discuss the induced anyonicity of a system of two identical planar charged particles without spin under the confining effect of a perpendicular magnetic field in a quantum dot. We show that this anyonic behaviour, inherent in quantum mechanics of bidimensional identical particles is due to the effects of Coulomb repulsion and that we observe characteristic aspects found exactly but existing in other treatments by different methods. r 2003 Elsevier B.V. All rights reserved. PACS: 03.65.Ge; 05.30.Pr; 71.10.Pm; 73.43.f Keywords: Anyon; Ground state; Strongly correlated particles 1. Introduction A considerable research effort has been devoted in recent years to the study of nanometric objects where the number of trapped particles can be precisely controlled. These objects, called quantum dots exhibit remarkable properties due to the strong correlations among the particles [1]. In most of the works, the Coulomb interaction between two electrons [2] is treated by first-order perturbation theory, the magnetic field being chosen strong enough to be considered as dominant effect. In this paper, we obtain analytically exact wavefunc- tions for two planar charged particles under a perpendi- cular magnetic field B and in a parabolic confining potential describing the quantum dots. 2. Exact wave functions In recent works [3–5], we have shown that the anyonic nature [6], appears to be induced by the competition between confining forces (magnetic field and confining potential) and repulsive forces (Coulomb potential). This competition may be appropriately described by a parameter C defined as 2a=l 0 ; the effective magnetic length (inducing the confining potential effect) over the Bohr radius. The relative Hamiltonian reads H r ¼ _ 2 2m ~ r r þ ieB 2_ ru f   2 þ 1 2 m o 2 0 r 2 þ e 2 4pee 0 1 ffiffi 2 p r ð1Þ expressed in cylindrical coordinates ðr; fÞ: Since the center of mass motion is a free motion, it is not useful to dwell on it. Following [3–5], the un-normalized wave function for a stationary state is of the form Fðr; fÞ¼ e i ðl þ xÞf ffiffiffiffiffi 2p p r 7jl þxj N n ðC 2 ; rÞe r 2 =4a 2 ; ð2Þ where N n ðC 2 ; xÞ is a special biconfluent Heun poly- nomial, l the angular momentum, x the real statistical parameter and a the effective confinement length. The condition to have a Heun polynomial leads us to two conditions of quantization: * energy quantization e ¼ e n ¼ n þ 1 þ s þ p with n ¼ 1; 2; y; N; * A nþ1 ¼ 0; which may be viewed as a new quantiza- tion rule. ARTICLE IN PRESS *Corresponding author. Tel.: +33-134-257089; fax: +33-1- 39-257004. E-mail address: arnaud.ralko@ptm.u-cergy.fr (A. Ralko). 0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.340