The specific topology of the line centered square lattice (known also as the Lieb lattice) induces remarkable spectral properties as the macroscopically degenerated zero energy flat band, the Dirac cone in the low energy spectrum, and the peculiar Hofstadter-type spectrum in magnetic field. We study here the properties of the finite Lieb lattice with periodic and vanishing boundary conditions. We find out the behavior of the flat band induced by disorder and external magnetic and electric fields. We show that in the confined Lieb plaquette threaded by a perpendicular magnetic flux there are edge states with nontrivial behavior. The specific class of twisted edge states, which have alternating chirality, are sensitive to disorder and do not support IQHE, but contribute to the longitudinal resistance. The symmetry of the transmittance matrix in the energy range where these states are located is revealed. The diamagnetic moments of the bulk and edge states in the Dirac-Landau domain, and also of the flat states in crossed magnetic and electric fields are shown.
We study Coulomb interacting electrons confined in polygonal quantum rings. We focus on the interplay of localization at the polygon corners and Coulomb repulsion. Remarkably, the Coulomb repulsion allows the formation of in-gap states, i.e., corner-localized states of electron pairs or clusters shifted to energies that were forbidden for non-interacting electrons, but below the energies of corner-side-localized states. We specify conditions allowing optical excitation to those states.
We calculate the orbital magnetization of single and double quantum dots coupled both by Coulomb interaction and by electron tunneling. The electronic states of the quantum dots are calculated in a tight-binding model, and the magnetization is discussed in relation to the energy spectrum and to the edge and bulk states. We identify effects of chirality of the electronic orbits and of the anticrossing of the energy levels when the magnetic field is varied. We also consider the effects of detuning the energy spectra of the quantum dots by an external gate potential. We compare our results with the recent experiments of Oosterkamp et al. ͓Phys. Rev. Lett. 80, 4951 ͑1998͔͒.
We investigate the transport properties of quantum dots placed in a strong magnetic field using a quantummechanical approach based on the two-dimensional tight-binding Hamiltonian with direct Coulomb interaction and the Landauer-Büttiker formalism. The electronic transmittance and the Hall resistance show Coulomb oscillations and also prove multiple addition processes. We identify this feature as the ''bunching'' of electrons observed in recent experiments and give an elementary explanation in terms of spectral characteristics of the dot. The spatial distribution of the added electrons may distinguish between the edge and bulk states and it has specific features for bunched electrons. The dependence of the charging energy on the number of electrons is discussed for a strong magnetic field. The crossover from the tunneling to quantum Hall regime is analyzed in terms of dot-lead coupling.
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