Quantum Hall Transition in the Classical Limit

We study the quantum Hall transition using the density-density correlation function. We show that in the limit h->0 the electron density moves along the percolating trajectories, undergoing normal diffusion. The localization exponent coincides with its percolation value \nu=4/3. The framework provides a natural way to study the renormalization group flow from percolation to quantum Hall transition. We also confirm numerically that the critical conductivity of a classical limit of quantum Hall transition is \sigma_{xx} = \sqrt{3}/4.Comment: 8 pages, 4 figures; substantial changes include the critical conductivity calculatio

Renormalizing Rectangles and Other Topics in Random Matrix Theory

We consider random Hermitian matrices made of complex or real $M\times N$ rectangular blocks, where the blocks are drawn from various ensembles. These matrices have $N$ pairs of opposite real nonvanishing eigenvalues, as well as $M-N$ zero eigenvalues (for $M>N$.) These zero eigenvalues are ``kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large $N,M$ limit, in which the ``rectangularity" $r={M\over N}$ is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large $N$ renormalization techniques. In addition to the kinematical $\delta$-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if $|r-1|$ is held fixed as $N\rightarrow\infty$, the $N$ non-zero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the non-zero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As $r\rightarrow 1$ the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that $r\rightarrow 1$ drives a cross over to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.Comment: LateX, 34 pages, 3 ps figure

Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials

Recently we introduced a family of $U(N)$ invariant Random Matrix Ensembles which is characterized by a parameter $\lambda$ describing logarithmic soft-confinement potentials $V(H) \sim [\ln H]^{(1+\lambda)} \:(\lambda>0$). We showed that we can study eigenvalue correlations of these "$\lambda$-ensembles" based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function $\exp[- (\ln x)^{1+\lambda}]$. In this work, we expand our previous work and show that: i) the eigenvalue density is given by a power-law of the form $\rho(x) \propto [\ln x]^{\lambda-1}/x$ and ii) the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called "ghost-correlation peak", is controlled by the parameter $\lambda$; decreasing $\lambda$ increases the anomaly. We also identify the two-level kernel of the $\lambda$-ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for $\lambda=1$. Finally, we discuss the universality of the $\lambda$-ensembles, which includes Wigner-Dyson universality ($\lambda \to \infty$ limit), the uncorrelated Poisson-like behavior ($\lambda \to 0$ limit), and a critical behavior for all the intermediate $\lambda$ ($0<\lambda<\infty$) in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the $N$ dependence of the two-level kernel of the fat-tail random matrices.Comment: 10 pages, 13 figure

Vibrational spectrum of topologically disordered systems

The topological nature of the disorder of glasses and supercooled liquids strongly affects their high-frequency dynamics. In order to understand its main features, we analytically studied a simple topologically disordered model, where the particles oscillate around randomly distributed centers, interacting through a generic pair potential. We present results of a resummation of the perturbative expansion in the inverse particle density for the dynamic structure factor and density of states. This gives accurate results for the range of densities found in real systems.Comment: Completely rewritten version, accepted in Physical Review Letter

Departures From Axisymmetric Morphology and Dynamics in Spiral Galaxies

New HI synthesis data have been obtained for six face-on galaxies with the Very Large Array. These data and reanalyses of three additional data sets make up a sample of nine face-on galaxies analyzed for deviations from axisymmetry in morphology and dynamics. This sample represents a subsample of galaxies already analyzed for morphological symmetry properties in the R-band. Four quantitative measures of dynamical nonaxisymmetry are compared to one another and to the quantitative measures of morphological asymmetry in HI and R-band to investigate the relationships between nonaxisymmetric morphology and dynamics. We find no significant relationship between asymmetric morphology and most of the dynamical measures in our sample. A possible relationship is found, however, between morphology and dynamical position angle differences between approaching and receding sides of the galaxy.Comment: 24 pages, 19 figures, AASTeX, accepted for publication in AJ, postscript figures available at ftp://culebra.tn.cornell.edu/pub/david/figures.tar.g

Black Hole Thermodynamics from Near-Horizon Conformal Quantum Mechanics

The thermodynamics of black holes is shown to be directly induced by their near-horizon conformal invariance. This behavior is exhibited using a scalar field as a probe of the black hole gravitational background, for a general class of metrics in D spacetime dimensions (with $D \geq 4$). The ensuing analysis is based on conformal quantum mechanics, within a hierarchical near-horizon expansion. In particular, the leading conformal behavior provides the correct quantum statistical properties for the Bekenstein-Hawking entropy, with the near-horizon physics governing the thermodynamic properties from the outset. Most importantly: (i) this treatment reveals the emergence of holographic properties; (ii) the conformal coupling parameter is shown to be related to the Hawking temperature; and (iii) Schwarzschild-like coordinates, despite their ``coordinate singularity,''can be used self-consistently to describe the thermodynamics of black holes.Comment: 16 pages. Sections 2 and 3 and sections 4 and 5 of version 1 were merged and reduced; a few typos were corrected. The original central results and equations remain unchange

A neutrino mass matrix with seesaw mechanism and two-loop mass splitting

We propose a model which uses the seesaw mechanism and the lepton number $\bar L = L_e - L_\mu - L_\tau$ to achieve the neutrino mass spectrum $m_1 = m_2$ and $m_3 = 0$, together with a lepton mixing matrix $U$ with $U_{e3} = 0$. In this way, we accommodate atmospheric neutrino oscillations. A small mass splitting $m_1 > m_2$ is generated by breaking $\bar L$ spontaneously and using Babu's two-loop mechanism. This allows us to incorporate ``just so'' solar-neutrino oscillations with maximal mixing into the model. The resulting mass matrix has three parameters only, since $\bar L$ breaking leads exclusively to a non-zero $ee$ matrix element.Comment: 8 pages, Late

Noncommutativity from spectral flow

We investigate the transition from second to first order systems. This transforms configuration space into phase space and hence introduces noncommutativity in the former. Quantum mechanically, the transition may be described in terms of spectral flow. Gaps in the energy or mass spectrum may become large which effectively truncates the available state space. Using both operator and path integral languages we explicitly discuss examples in quantum mechanics, (light-front) quantum field theory and string theory.Comment: 31 pages, one Postscript figur

Radiatively Induced Neutrino Masses and Oscillations in an SU(3)_LxU(1)_N Gauge Model

We have constructed an $SU(3)_L \times U(1)_N$ gauge model utilizing an $U(1)_{L^\prime}$ symmetry, where $L^\prime$ = $L_e-L_\mu-L_\tau$, which accommodates tiny neutrino masses generated by $L^\prime$-conserving one-loop and $L^\prime$-breaking two-loop radiative mechanisms. The generic smallness of two-loop radiative effects compared with one-loop radiative effects describes the observed hierarchy of $\Delta m_{atm}^2$ $\gg$ $\Delta m_\odot^2$. A key ingredient for radiative mechanisms is a charged scalar ($h^+$) that couples to charged lepton-neutrino pairs and $h^+$ together with the standard Higgs scalar ($\phi$) can be unified into a Higgs triplet as ($\phi^0$, $\phi^-$, $h^+$)$^T$. This assignment in turn requires lepton triplets ($\psi_L^i$) with heavy charged leptons ($\kappa_L^{+i}$) as the third member: $\psi_L^i=(\nu^i_L,\ell^i_L,\kappa^{+i}_L)^T$, where $i$ ($=1,2,3$) denotes three families. It is found that our model is relevant to yield quasi-vacuum oscillations for solar neutrinos.Comment: 11 pages, revtex, including 2 figures, accepted for publication in Phys. Rev. D with minor modification of our resul

Entropy: From Black Holes to Ordinary Systems

Several results of black holes thermodynamics can be considered as firmly founded and formulated in a very general manner. From this starting point we analyse in which way these results may give us the opportunity to gain a better understanding in the thermodynamics of ordinary systems for which a pre-relativistic description is sufficient. First, we investigated the possibility to introduce an alternative definition of the entropy basically related to a local definition of the order in a spacetime model rather than a counting of microstates. We show that such an alternative approach exists and leads to the traditional results provided an equilibrium condition is assumed. This condition introduces a relation between a time interval and the reverse of the temperature. We show that such a relation extensively used in the black hole theory, mainly as a mathematical trick, has a very general and physical meaning here; in particular its derivation is not related to the existence of a canonical density matrix. Our dynamical approach of thermodynamic equilibrium allows us to establish a relation between action and entropy and we show that an identical relation exists in the case of black holes. The derivation of such a relation seems impossible in the Gibbs ensemble approach of statistical thermodynamics. From these results we suggest that the definition of entropy in terms of order in spacetime should be more general that the Boltzmann one based on a counting of microstates. Finally we point out that these results are obtained by reversing the traditional route going from the Schr\"{o}dinger equation to statistical thermodynamics