An Infinite Dimensional Approach to the Third Fundamental Theorem of Lie

We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Chiral-Yang-Mills theory, non commutative differential geometry, and the need for a Lie super-algebra

In Yang-Mills theory, the charges of the left and right massless Fermions are independent of each other. We propose a new paradigm where we remove this freedom and densify the algebraic structure of Yang-Mills theory by integrating the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions of opposite chiralities. Using the Bianchi identity, we prove that the corresponding covariant differential is associative if and only if we gauge a Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally occurs along an odd generator of the super-algebra and induces a representation of the Connes-Lott non commutative differential geometry of the 2-point finite space.Comment: 17 pages, no figur

A New Pseudopolymorph of Hexakis-(4-cynaophenyl)benzene

The title compound (systematic name: benzene-4,4′,4′′,4′′′,-4′′′′,4′′′′′-hexaylhexabenzonitrile dichloromethane disolvate), C48H24N6•2CH2Cl2, crystallizes as an inclusion compound during the slow diffusion of methanol into a solution of hexakis(4-cyanophenyl)benzene in CH2Cl2. The hexakis(4- cyanophenyl)benzene molecule lies on an axis of twofold rotation in the space group Pbcn. Weak C—H•••N interactions between hexakis(4-cyanophenyl)benzene molecules define an open network with space for including guests. The resulting structure is a new pseudopolymorph of hexakis-(4-cyanophenyl)benzene. The eight known pseudopolymorphs have few shared architectural features, in part because none of the intermolecular interactions that are present plays a dominant role or forces neighboring molecules to assume particular relative orientations

Electric charge enhancements in carbon nanotubes: Theory and experiments

We present a detailed study of the static enhancement effects of electric charges in micron-length single-walled carbon nanotubes, using theoretically an atomic charge-dipole model and experimentally electrostatic force microscopy. We demonstrate that nanotubes exhibit at their ends surprisingly weak charge enhancements which decrease with the nanotube length and increase with the nanotube radius. A quantitative agreement is obtained between theory and experiments.Comment: 6 Fi

On transport in quantum Hall systems with constrictions

Motivated by recent experimental findings, we study transport in a simple phenomenological model of a quantum Hall edge system with a gate-voltage controlled constriction lowering the local filling factor. The current backscattered from the constriction is seen to arise from the matching of the properties of the edge-current excitations in the constriction ($\nu_{2}$) and bulk ($\nu_{1}$) regions. We develop a hydrodynamic theory for bosonic edge modes inspired by this model, finding that a competition between two tunneling process, related by a quasiparticle-quasihole symmetry, determines the fate of the low-bias transmission conductance. In this way, we find satisfactory explanations for many recent puzzling experimental results.Comment: 4 pages, 4 figure

Exact wavefunctions for excitations of the nu=1/3 fractional quantum Hall state from a model Hamiltonian

We study fractional quantum Hall states in the cylinder geometry with open boundaries. By truncating the Coulomb interactions between electrons we show that it is possible to construct infinitely many exact eigenstates including the ground state, quasiholes, quasielectrons and the magnetoroton branch of excited states.Comment: 7 pages, 3 figures, longer published versio

Composite-fermionization of bosons in rapidly rotating atomic traps

The non-perturbative effect of interaction can sometimes make interacting bosons behave as though they were free fermions. The system of neutral bosons in a rapidly rotating atomic trap is equivalent to charged bosons coupled to a magnetic field, which has opened up the possibility of fractional quantum Hall effect for bosons interacting with a short range interaction. Motivated by the composite fermion theory of the fractional Hall effect of electrons, we test the idea that the interacting bosons map into non-interacting spinless fermions carrying one vortex each, by comparing wave functions incorporating this physics with exact wave functions available for systems containing up to 12 bosons. We study here the analogy between interacting bosons at filling factors $\nu=n/(n+1)$ with non-interacting fermions at $\nu^*=n$ for the ground state as well as the low-energy excited states and find that it provides a good account of the behavior for small $n$, but interactions between fermions become increasingly important with $n$. At $\nu=1$, which is obtained in the limit $n\rightarrow \infty$, the fermionization appears to overcompensate for the repulsive interaction between bosons, producing an {\em attractive} interactions between fermions, as evidenced by a pairing of fermions here.Comment: 8 pages, 3 figures. Submitted to Phys. Rev.

Spectroscopy for cold atom gases in periodically phase-modulated optical lattices

The response of cold atom gases to small periodic phase modulation of an optical lattice is discussed. For bosonic gases, the energy absorption rate is given, within linear response theory, by imaginary part of the current correlation function. For fermionic gases in a strong lattice potential, the same correlation function can be probed via the production rate double occupancy. The phase modulation gives thus direct access to the conductivity of the system, as function of the modulation frequency. We give an example of application in the case of one dimensional bosons at zero temperature and discuss the link between the phase- and amplitude-modulation.Comment: 4 pages, 2 figures, final versio

How do random Fibonacci sequences grow?

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by $$\int_0^\infty \log x d\nu_\alpha (x),$$ where $\alpha$ is an explicit function of $p$ depending on the case we consider, and $\nu_\alpha$ is an explicit probability distribution on \RR_+ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<p\le1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative