Quantum Hall Transition in an Array of Quantum Dots

A two-dimensional array of quantum dots in a magnetic field is considered. The electrons in the quantum dots are described as unitary random matrix ensembles. The strength of the magnetic field is such that there is half a flux quantum per plaquette. This model exhibits the Integer Quantum Hall Effect. For N electronic states per quantum dot the limit N → ∞ can be solved by a saddle point integration of a supersymmetric field theory. The effect of level statistics on the density of states and the Hall conductivity is compared with the effect of temperature fluctuations. PACS Nos.: 71.55Jv, 73.20Dx, 73.40Hm We consider a two-dimensional array of quantum dots in a homogeneous magnetic field perpendicular to the array. A quantum dot in an array is a complex finite system of electrons, subject to strong Coulomb interaction and a confining potential. Even if the number of electrons is small there is a large number of electronic states in a given energy interval. Therefore, we are forced to use a statistical description of the quantum dot. A typical feature of such a complicated non-integrable system is level repulsion. The latter, also found in other complex many-particle systems like atomic nuclei [1], atoms [2] or metallic particles [3], can be conveniently described by random matrix ensembles [4]. Since the magnetic field breaks the time–reversal invariance in the dot, an appropriate model is the Gaussian unitary ensemble (GUE). Electrons can travel in the array of quantum dots due to tunneling between neighboring dots. On the square–array, which will be considered in this article, the tunneling rates are t and t for nearest and next nearest neighbors, respectively (cf. Fig.1). The coupling between the individual quantum dots due to these tunneling processes is weak. This allows us to assume that the statistical occupation of the electronic states in each dot is uncorrelated between different dots. Thus the quantum dots