Generalized Hofstadter picture for lattices with more than one degree of freedom per site

Starting from a transfer-matrix formalism, we study the electronic structure of a model of a periodic two-dimensional array of small disks of electrons or ‘‘quantum dots’’ that is infinite in both the x and y directions and is in a transverse magnetic field. The disks are coupled via scattering matrices which allow electrons to be transmitted between them. Unlike tight-binding models, such systems must in general allow for more than one degree freedom per site. Despite this, the spectrum for this system, which is dependent on the transmission probability, has many properties in common with spectra that have been obtained for periodic tight-binding systems. In particular, the energy bands become split when the magnetic field takes rational values and the current amplitudes can obey a type of Harper’s equation. In certain limits, the spectrum generated by our model becomes equivalent to that of the standard two-dimensional tight-binding model.