Theory of the Hall effect in two-dimensional quantum-dot arrays

We investigate the problem of lateral ballistic quantum transport in two-dimensional periodic arrays of quantum dots in a transverse magnetic field. We argue that the Büttiker equations, applied to an array of quantum dots attached to ideal reservoirs, produce a Hall conductance which may be either positive or negative, and quantized in integer or fractional multiples of ${\mathit{e}}^2$/h. The fractions differ in value and in origin from the usual fractional quantum Hall effect. The physical reason for the complexity of this transport problem is that a quantum-dot array is capable of supporting edge currents which rotate either in a normal (right-hand) sense in a magnetic field, or in the opposite direction, or which are superpositions of different numbers of independent normal and counter-rotating states. The edge-state spectrum is therefore very rich, and a rich and interesting variety of transport phenomena is demonstrated. The effect of disorder on the transport problem (nonballistic transport) is briefly discussed.