The Absence of the Fractional Quantum Hall Effect at High Landau Levels

We compare the energies of the Laughlin liquid and a charge density wave in a weak magnetic field for the upper Landau level filling factors $\nu_N = 1/3$ and $1/5$. The charge density wave period has been optimized and was found to be $\simeq 3R_c$, where $R_c$ is the cyclotron radius. We conclude that the optimal charge density wave is more energetically preferable than the Laughlin liquid for the Landau level numbers $N \ge 2$ at $\nu_N = 1/3$ and for $N \ge 3$ at $\nu_N = 1/5$. This implies that the $1/3$ fractional quantum Hall effect cannot be observed for $N \ge 2$, in agreement with the experiment.Comment: 12 pages, revtex, 2 PostScript figures are applied. Revised and corrected version. Also available at http://www.mnhep.umn.edu/~mfogler

Charging Spectrum of a Small Wigner Crystal Island

Charging of a clean two-dimensional island is studied in the regime of small concentration of electrons when they form the Wigner crystal. The number of electrons in the island is assumed to be not too big (N < 100). It is shown that the total energy of the island as a function of N has a quasi-periodic component of a universal shape, that is independent of the form of electron-electron interactions. These oscillations are caused by the combination of the geometric effects associated with packing of the triangular lattice into the circular island. These effects are: the shell effect, associated with starting a new crystalline row, and the so-called confinement polaronic effect. In the presence of close metallic gates, which eliminate the long-range part of the electron-electron interactions, the oscillations of the energy bring about simultaneous entering of the dot by a few electrons.Comment: 8 pages, Latex, 8 Postscript pages are include

Sperry versus Hebb: topographic mapping in Isl2/EphA3 mutant mice

BACKGROUND: In wild-type mice, axons of retinal ganglion cells establish topographically precise projection to the superior colliculus of the midbrain. This means that axons of neighboring retinal ganglion cells project to the proximal locations in the target. The precision of topographic projection is a result of combined effects of molecular labels, such as Eph receptors and ephrins, and correlated neural activity. In the Isl2/EphA3 mutant mice the expression levels of molecular labels are changed. As a result the topographic projection is rewired so that the neighborhood relationships between retinal cell axons are disrupted. RESULTS: Here we study the computational model for retinocollicular connectivity formation that combines the effects of molecular labels and correlated neural activity. We argue that the effects of correlated activity presenting themselves in the form of Hebbian learning rules can facilitate the restoration of the topographic connectivity even when the molecular labels carry conflicting instructions. This occurs because the correlations in electric activity carry information about retinal cells' origin that is independent on molecular labels. We argue therefore that partial restoration of the topographic property of the retinocollicular projection observed in Isl2/EphA3 heterozygous knockin mice may be explained by the effects of correlated neural activity. We address the maps observed in Isl2/EphA3 knockin/EphA4 knockout mice in which the levels of retinal labels are uniformly reduced. These maps can be explained by either the saturation of EphA receptor mapping leading to the relative signaling model or by the reverse signaling conveyed by ephrin-As expressed by retinal axons. CONCLUSION: According to our model, experiments in Isl2/EphA3 knock-in mice test the interactions between effects of molecular labels and correlated activity during the development of neural connectivity. Correlated activity can partially restore topographic order even when molecular labels carry conflicting information