  The ground state persistent current and electron addition spectrum in
two-dimensional quantum dot arrays and one-dimensional quantum dot rings,
pierced by an external magnetic flux, are investigated using the extended
Hubbard model. The collective multidot problem is shown to map exactly into the
strong field noninteracting finite-size Hofstadter butterfly problem at the
spin polarization transition. The finite size Hofstadter problem is discussed,
and an analytical solution for limiting values of flux is obtained. In weak
fields we predict novel flux periodic oscillations in the spin component along
the quantization axis with a periodicity given by $\nu h/e$ ($\nu \le 1$). The
sensitivity of the calculated persistent current to interaction and disorder is
shown to reflect the intricacies of various Mott-Hubbard quantum phase
transitions in two- dimensional systems: the persistent current is suppressed
in the antiferromagnetic Mott-insulating phase governed by intradot Coulomb
interactions; the persistent current is maximized at the spin density wave -
charge density wave transition driven by the nearest neighbor interdot
interaction; the Mott-insulating phase persistent current is enhanced by the
long-range interdot interactions to its noninteracting value; the strong
suppression of the noninteracting current in the presence of random disorder is
seen only at large disorder strengths; at half filling even a relatively weak
intradot Coulomb interaction enhances the disordered noninteracting system
persistent current; in general, the suppression of the persistent current by
disorder is less significant in the presence of the long-range interdot Coulomb
interaction.
