Selected Topics in Graphene physics

Graphene research is currently one of the largest fields in condensed matter. Due to its unusual electronic spectrum with Dirac-like quasiparticles, and the fact that it is a unique example of a metallic membrane, graphene has properties that have no match in standard solid state textbooks. In these lecture notes, I discuss some of these properties that are not covered in detail in recent reviews. We study the particular aspects of the physics/chemistry of carbon that influence the properties of graphene; the basic features of graphene's band structure including the pi and sigma bands; the phonon spectra in free floating graphene; the effects of a substrate on the structural properties of graphene; and the effect of deformations in the propagation of electrons. The objective of these notes is not to provide an unabridged theoretical description of graphene but to point out some of the peculiar aspects of this material.Comment: 19 pages, 13 figures. Lecture notes for the Les Houches School on "Modern theories of correlated electron systems", May 11-29, 2009

Pairing, Stripes, Lattice Distortions and Superconductivity in Cuprate Oxides

We propose a model for a spatially modulated collective state of superconducting cuprates in which the electronic properties vary locally in space. In this model the regions of higher hole density (called stripes) are described as Luttinger liquids and the regions of lower density (antiferromagnetic ladders) as an interacting bosonic gas of d$_{x^2-y^2}$ hole pairs. We show that the transition to the superconducting state is topological and driven by decay processes among these elementary excitations in the presence of vibrations.Comment: Proceedings of the MTSC 2000, Klosters, Switzerland, April 1, 200

Limits of Tangents of a Quasi-Ordinary Hypersurface

We compute explicitly the limits of tangents of a quasi-ordinary singularity in terms of its special monomials. We show that the set of limits of tangents of Y is essentially a topological invariant of Y .Comment: 10 pages

Desingularization of Legendrian Surfaces

In this paper we prove a desingularization theorem for Legendrian surfaces that are the conormal of a quasi-ordinary hypersurface

How big was Galileo's impact? Percussion in the Sixth Day of the "Two New Sciences"

The Giornata Sesta about the Force of Percussion is a relatively less known Chapter from the Galileo's masterpiece "Discourse about Two New Sciences". It was first published lately (1718), long after the first edition of the Two New Sciences (1638) and Galileo's death (1642). The Giornata Sesta focuses on how to quantify the percussion force caused by a body in movement, and describes a very interesting experiment known as "the two-bucket experiment". In this paper, we review this experiment reported by Galileo, develop a steady-state theoretical model, and solve its transient form numerically; additionally, we report the results from one real simplified analogous experiment. Finally, we discuss the conclusions drawn by Galileo -- correct, despite a probably unnoticeable imbalance --, showing that he did not report the thrust force component in his setup -- which would be fundamental for the correct calculation of the percussion force.Comment: 9 pages, 2 figure

Localized Magnetic States in 2D Semiconductors

We study the formation of magnetic states in localized impurities embedded into two-dimensional semiconductors. By considering various energy configurations, we illustrate the interplay of the gap and the bands in the system magnetization. Finally, we consider finite-temperature effects to show how increasing $T$ can lead to formation and destruction of magnetization.Comment: 5 pages, 4 figure

Non-Fermi Liquid Behavior in U and Ce Alloys: Criticality, Disorder, Dissipation, and Griffiths-McCoy singularities

In this paper we provide the theoretical basis for the problem of Griffiths-McCoy singularities close to the quantum critical point for magnetic ordering in U and Ce intermetallics. We show that the competition between Kondo effect and RKKY interaction can be expressed in Hamiltonian form and the dilution effect due to alloying leads to a quantum percolation problem driven by the number of magnetically compensated moments. We argue that the exhaustion paradox proposed by Nozi\`eres is explained when the RKKY interaction is taken into account. We revisited the one impurity and two impurity Kondo problem and showed that in the presence of particle-hole symmetry breaking operators the system flows to a line of fixed points characterized by coherent motion of the spins. This leads to a {\it cluster Kondo effect}. We calculate explicitly from the microscopic Hamiltonian the parameters which appear in all the response functions. We show that there is a maximum number $N_c$ of spins in the clusters such that above this number tunneling ceases to occur. These effects lead to a distribution of cluster Kondo temperatures which vanishes for finite clusters and therefore leads to strong magnetic response. From these results we propose a {\it dissipative quantum droplet model} which describes the critical behavior of metallic magnetic systems. This model predicts that in the paramagnetic phase there is a crossover temperature $T^*$ above which Griffiths-McCoy like singularities with power law behavior. Below $T^*$, however, a new regime dominated by dissipation occurs with stronger divergences. We estimate that $T^*$ is exponentially small with $N_c$.Comment: 97 pages, 8 postscript figue

Master Equation for a Particle Coupled to a Two-Level Reservoir

We study the quantum dissipative dynamics of a particle coupled linearly to a set of two-level systems (the heat bath) via the master equation method which we extract from the path integral formalism independently from the form of the bath spectral density. We compare our results with the standard models based on bosonic heat baths showing their main differences and similarities. In particular, we study special forms for the spectral density of the bath which give results quite different from the standard models.Comment: 11 pages, RevTeX 3.

Droplets in Disordered Metallic Quantum Critical Systems

We present a field theory for a structurally disordered magnetic system coupled to a metallic environment near a quantum critical point. We show that close to the magnetic quantum critical point droplets are formed due to the disorder and undergo dissipative quantum dynamics. We show that the problem has a characteristic energy scale, the droplet Kondo temperature, that determines the crossover energy scale from weak to strong coupling. Our results have direct significance for the Griffiths-McCoy singularities of itinerant magnets.Comment: 4 Pages, 2 Figure

The mesoscopic magnetron as an open quantum system

Motivated by the emergence of materials with mean free paths on the order of microns, we propose a novel class of solid state radiation sources based on reimplementing classical vacuum tube designs in semiconductors. Using materials with small effective masses, these devices should be able to access the terahertz range. We analyze the DC and AC operation of the simplest such device, the cylindrical diode magnetron, using effective quantum models. By treating the magnetron as an open quantum system, we show that it continues to operate as a radiation source even if its diameter is only a few tens of magnetic lengths.Comment: 11 pages, 7 figures; submitted to Physical Review Applie