Lattice gauge theory model for graphene

The effects of the electromagnetic (e.m.) electron-electron interactions in half-filled graphene are investigated in terms of a lattice gauge theory model. By using exact Renormalization Group methods and lattice Ward Identities, we show that the e.m. interactions amplify the responses to the excitonic pairings associated to a Kekul\'e distortion and to a charge density wave. The effect of the electronic repulsion on the Peierls-Kekul\'e instability, usually neglected, is evaluated by deriving an exact non-BCS gap equation, from which we find evidence that strong e.m. interactions among electrons facilitate the spontaneous distortion of the lattice and the opening of a gap.Comment: 5 pages, 2 figures; typos corrected. Final version published in Phys. Rev.

Absence of interaction corrections in graphene conductivity

The exact vanishing of the interaction corrections to the zero temperature and zero frequency conductivity of graphene in the presence of weak short range interactions is rigorously established.Comment: 4 page

Microhandling devices for the assembly of Hybrid Microproducts

Hybrid microproducts are very important in every application where small dimension and light weigh are fundamental. They are used in different fields, such as the automotive, the aeronautic/aerospace, the automation, the medical and biomedical field, the watch industry, but also in the sport and entertainment sectors. These microproducts differ from MOEMS because they consist of several components with some features of few hundreds microns made of different materials. The final product is obtained by the assembly of the various components and this gives hybrid microproducts a high 3D aspect and better mechanical performance than MOEMS. The main problem that limits the market diffusion of these products is the enormous assembly cost that represents up to the 80% of their final cost. This is due to their manual assembly that is a time consuming activity. Actually, traditional assembly systems are not suitable because when dimensions of objects are less than 1 mm, many problems arise: surface forces become bigger than gravity, there are difficulties due to the vision and force control, objects can be fragile, tolerances become very narrow, there is not a systematic design for microassembly, methods and models are not developed enough in microdomain. Different approaches can be found in literature as an alternative to manual assembly to reduce costs. One of the most promising strategies is the assembly microfactory. It is a miniaturization of an assembly system with the aims of improving precision and reducing the ratio, in terms of energy, material and space consumption, between the assembly facilities and the microproducts obtained. Therefore, many assembly and control devices have to work in a very little area with low energy consumption and high precision. Nowadays, one of the main problems that makes difficult the development of these microfactories is the lack of handling devices able to manipulate components that have dimension of few hundreds microns. The required handling devices are microfeeders to transport, orient and position microparts, microsorters to arrange components, microgrippers to grasp and release them, micromanipulators and microrobots to support microgrippers. In the PhD research activity various innovative handling devices for manipulating parts of different materials and shapes have been designed, developed and tested. These systems are able to solve some critical issues in the automatic handling of objects with dimensions of few hundreds microns such as feeding and transporting, grasping and releasing, positioning. The developed systems are an electrostatic centering device and an electrostatic sorter, some mechanical and adhesive grippers and a 2 DOF microrobot. The two electrostatic devices make use of electrostatic force to transport and position microparts (the centering device) and to sort and arrange a group of objects randomly positioned (the sorter system). Two kinds of gripper have been designed: the mechanical (piezo actuated) and the adhesive gripper. The first kind uses a piezoelectric actuator to close and open the fingers, the second exploits capillary forces to grasp and center microobjects. Finally, the 2 DOF manipulator is an electromagnetic device able to support light grippers as electrostatic and adhesive ones. These devices have been integrated in the assembly microfactory in progress at the Department of Mechanical, Nuclear and Production Engineering of the University of Pisa within the Italian research PRIN project “Development of innovative technologies for the assembly of hybrid microproducts”. In this microfactory, the assembly strategy makes use of a carrier that contains the components of the products to be assembled and moves from an assembly station to the next one. This assembly system is confined in a controlled environment and consists, at the moment, of an assembly station equipped with various grippers (on the basis of the assembly tasks to be carried out) and controlled by vision systems

Multi-channel Luttinger Liquids at the Edge of Quantum Hall Systems

We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic 2d theory

Multi-channel Luttinger Liquids at the Edge of Quantum Hall Systems

We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic 2d theory

Borel summability of ϕ44\phi^{4}_{4} planar theory via multiscale analysis

We review the issue of Borel summability in the framework of multiscale analysis and renormalization group, by discussing a proof of Borel summability of the $\phi^{4}_4$ massive euclidean planar theory; this result is not new, since it was obtained by Rivasseau and 't Hooft. However, the techniques that we use have already been proved effective in the analysis of various models of consended matter and field theory; therefore, we take the $\phi^{4}_4$ planar theory as a toy model for future applications.Comment: 30 pages, 8 figures. Notations revised, and other minor change

Dynamics of mean-field Fermi systems with nonzero pairing

We study the dynamics of many-body Fermi systems, for a class of initial data which are close to quasi-free states exhibiting a nonvanishing pairing matrix. We focus on the mean-field scaling, which for fermionic systems is naturally coupled with a semiclassical scaling. Under the assumption that the initial datum enjoys a suitable semiclassical structure, we give a rigorous derivation of the time-dependent Hartree-Fock-Bogoliubov equation, a nonlinear effective evolution equation for the one-particle density matrix of the system, as the number of particles goes to infinity. Our result holds for all macroscopic times, and provides bounds for the rate of convergence.Comment: 43 page

Response functions of many-body condensed matter systems

We discuss rigorous results about the response functions of gapped and gapless interacting fermionic lattice models, relevant for the study of transport in condensed matter physics.Comment: 27 pages. Article commissioned by the Encyclopedia of Mathematical Physic

Effective Dynamics of Extended Fermi Gases in the High-Density Regime

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases