Arithmetic-Progression-Weighted Subsequence Sums

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\odot S|\geq \min\{|G|-1,\,n\}$, that $W\odot S=G$ if $n\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\odot S\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\equiv \alpha\mod n,$$ where $\alpha,a_1,..., a_r\in \Z$ are given, has a solution $(x_1,...,x_r)\in \Z^r$ modulo $n$ with all $x_i$ distinct modulo $n$. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G\cong C_{n_1}\oplus C_{n_2}$ (where $n_1\mid n_2$ and $n_2\geq 3$) having $k$ distinct terms, for any $k\in [3,\min\{n_1+1,\,\exp(G)\}]$. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence

Coulomb drag between one-dimensional conductors

We have analyzed Coulomb drag between currents of interacting electrons in two parallel one-dimensional conductors of finite length $L$ attached to external reservoirs. For strong coupling, the relative fluctuations of electron density in the conductors acquire energy gap $M$. At energies larger than $\Gamma = const \times v_- \exp (-LM/v_-)/L + \Gamma_{+}$, where $\Gamma_{+}$ is the impurity scattering rate, and for $L>v_-/M$, where $v_-$ is the fluctuation velocity, the gap leads to an ``ideal'' drag with almost equal currents in the conductors. At low energies the drag is suppressed by coherent instanton tunneling, and the zero-temperature transconductance vanishes, indicating the Fermi liquid behavior.Comment: 5 twocolumn pages in RevTex, added 1 eps-Figure and calculation of trans-resistanc

Strong-coupling branching of FQHL edges

We have developed a theory of quasiparticle backscattering in a system of point contacts formed between single-mode edges of several Fractional Quantum Hall Liquids (FQHLs) with in general different filling factors $\nu_j$ and one common single-mode edge $\nu_0$ of another FQHL. In the strong-tunneling limit, the model of quasiparticle backscattering is obtained by the duality transformation of the electron tunneling model. The new physics introduced by the multi-point-contact geometry of the system is coherent splitting of backscattered quasiparticles at the point contacts in the course of propagation along the common edge $\nu_0$. The ``branching ratios'' characterizing the splitting determine the charge and exchange statistics of the edge quasiparticles that can be different from those of Laughlin's quasiparticles in the bulk of FQHLs. Accounting for the edge statistics is essential for the system of more than one point contact and requires the proper description of the flux attachement to tunneling electrons.Comment: 12 pages, 2 figure

Fractional charge in transport through a 1D correlated insulator of finite length

Transport through a one channel wire of length $L$ confined between two leads is examined when the 1D electron system has an energy gap $2M$: $M > T_L \equiv v_c/L$ induced by the interaction in charge mode ($v_c$: charge velocity in the wire). In spinless case the transformation of the leads electrons into the charge density wave solitons of fractional charge $q$ entails a non-trivial low energy crossover from the Fermi liquid behavior below the crossover energy $T_x \propto \sqrt{T_L M} e^{-M /[T_L(1-q^2)]}$ to the insulator one with the fractional charge in current vs. voltage, conductance vs. temperature, and in shot noise. Similar behavior is predicted for the Mott insulator of filling factor $\nu = integer/(2 m')$.Comment: 5 twocolumn pages in RevTex, no figure

Quantum-Hall activation gaps in graphene

We have measured the quantum-Hall activation gaps in graphene at filling factors $\nu=2$ and $\nu=6$ for magnetic fields up to 32 T and temperatures from 4 K to 300 K. The $\nu =6$ gap can be described by thermal excitation to broadened Landau levels with a width of 400 K. In contrast, the gap measured at $\nu=2$ is strongly temperature and field dependent and approaches the expected value for sharp Landau levels for fields $B > 20$ T and temperatures $T > 100$ K. We explain this surprising behavior by a narrowing of the lowest Landau level.Comment: 4 pages, 4 figures, updated version after review, accepted for PR

Scaling of the quantum-Hall plateau-plateau transition in graphene

The temperature dependence of the magneto-conductivity in graphene shows that the widths of the longitudinal conductivity peaks, for the N=1 Landau level of electrons and holes, display a power-law behavior following $\Delta \nu \propto T^{\kappa}$ with a scaling exponent $\kappa = 0.37\pm0.05$. Similarly the maximum derivative of the quantum Hall plateau transitions $(d\sigma_{xy}/d\nu)^{max}$ scales as $T^{-\kappa}$ with a scaling exponent $\kappa = 0.41\pm0.04$ for both the first and second electron and hole Landau level. These results confirm the universality of a critical scaling exponent. In the zeroth Landau level, however, the width and derivative are essentially temperature independent, which we explain by a temperature independent intrinsic length that obscures the expected universal scaling behavior of the zeroth Landau level

Threshold features in transport through a 1D constriction

Suppression of electron current $\Delta I$ through a 1D channel of length $L$ connecting two Fermi liquid reservoirs is studied taking into account the Umklapp electron-electron interaction induced by a periodic potential. This interaction causes Hubbard gaps $E_H$ for $L \to \infty$. In the perturbative regime where $E_H \ll v_c/L$ ($v_c:$ charge velocity), and for small deviations $\delta n$ of the electron density from its commensurate values $- \Delta I/V$ can diverge with some exponent as voltage or temperature $V,T$ decreases above $E_c=max(v_c/L,v_c \delta n)$, while it goes to zero below $E_c$. This results in a nonmonotonous behavior of the conductance.Comment: Final variant published in PRL, 79, 1714; minor correction

Gap opening in the zeroth Landau level of graphene

We have measured a strong increase of the low-temperature resistivity $\rho_{xx}$ and a zero-value plateau in the Hall conductivity $\sigma_{xy}$ at the charge neutrality point in graphene subjected to high magnetic fields up to 30 T. We explain our results by a simple model involving a field dependent splitting of the lowest Landau level of the order of a few Kelvin, as extracted from activated transport measurements. The model reproduces both the increase in $\rho_{xx}$ and the anomalous $\nu=0$ plateau in $\sigma_{xy}$ in terms of coexisting electrons and holes in the same spin-split zero-energy Landau level.Comment: 4 pages, 3 figure