Eric Akkermans

VORTICES IN GINZBURG-LANDAU BILLIARDS

We present an analysis of the Ginzburg-Landau equations for the description of a two-dimensional superconductor in a bounded domain. Using the properties of a particular integrability point of these equations which allows vortex solutions, we obtain a closed expression for the energy of the superconductor. The role of the boundary of the system is to provide a selection mechanism for the number of vortices. A geometrical interpretation of these results is presented and they are applied to the analysis of the magnetization recently measured on small superconducting discs. Problems related to the interaction and nucleation of vortices are discussed.

Revealing the Topology of Quasicrystals with a Diffraction Experiment

Topological properties of crystals and quasicrystals is a subject of recent and growing interest. This Letter reports an experiment where, for certain quasicrystals, these properties can be directly retrieved from diffraction. We directly observe, using an interferometric approach, all of the topological invariants of finite-length Fibonacci chains in their diffraction pattern. We also quantitatively demonstrate the stability of these topological invariants with respect to structural disorder.

Observing a scale anomaly and a universal quantum phase transition in graphene

One of the most interesting predictions resulting from quantum physics, is the violation of classical symmetries, collectively referred to as anomalies. A remarkable class of anomalies occurs when the continuous scale symmetry of a scale-free quantum system is broken into a discrete scale symmetry for a critical value of a control parameter. This is an example of a (zero temperature) quantum phase transition. Such an anomaly takes place for the quantum inverse square potential known to describe ‘Efimov physics’. Broken continuous scale symmetry into discrete scale symmetry also appears for a charged and massless Dirac fermion in an attractive 1/r Coulomb potential. The purpose of this article is to demonstrate the universality of this quantum phase transition and to present convincing experimental evidence of its existence for a charged and massless fermion in an attractive Coulomb potential as realized in graphene.When the continuous scale symmetry of a quantum system is broken, anomalies occur which may lead to quantum phase transitions. Here, the authors provide evidence for such a quantum phase transition in the attractive Coulomb potential of vacancies in graphene, and further envision its universality for diverse physical systems.

Mesoscopic physics of photons

We review the general features of coherent multiple scattering of electromagnetic waves in random media. In particular, coherent backscattering and angular correlation functions of speckle patterns are studied in some detail. We present a general formalism based on a physically intuitive description that also permits us to derive quantitative expressions. Then, the notion of phase boxes describing the quantum crossings of diffusons is discussed. This notion permits us to understand the long-range correlations that are at the origin of most of the mesoscopic effects either for electrons or photons. Then, we turn to the problem of decoherence, namely, the washing out of interference effects. We use as an example the effect of a nondeterministic motion of the scatterers. We discuss some applications of these ideas to diffusive wave spectroscopy, including calculations of the intensity–time correlation function in the presence of quantum crossings.

Coherent Multiple Scattering in Disordered Media

These notes contain a rapid overview of the methods and results obtained in the field of propagation of waves in disordered media. The case of Schrodinger and Helmholtz equations are considered that describe respectively electrons in metals and scalar electromagnetic waves. The assumptions on the nature of disorder are discussed and perturbation methods in the weak disorder limit are presented. A central quantity, namely the probability of quantum diffusion is defined and calculated in the same limit. It is then shown that several relevant physical quantities are related to the return probability. Examples are provided to substantiate this, which include the average electrical conductivity, its fluctuations, the average albedo and spectral correlations.

Scale anomaly of a Lifshitz scalar: a universal quantum phase transition to discrete scale invariance

We demonstrate the existence of a universal transition from a continuous scale invariant phase to a discrete scale invariant phase for a class of one-dimensional quantum systems with anisotropic scaling symmetry between space and time. These systems describe a Lifshitz scalar interacting with a background potential. The transition occurs at a critical coupling

Coherent Effects in the Multiple Scattering of Light in Random Media

lambda_{c}

Geometrical Description of Vortices in Ginzburg-Landau Billiards

corresponding to a strongly attractive potential.

HEAT KERNEL OF INTEGRABLE BILLIARDS IN A MAGNETIC FIELD

We review some of the characteristic features of the coherent multiple scattering of scalar electromagnetic waves in random media. The probability of quantum diffusion is defined and calculated up to the contribution of the cooperon. We show that there are additional corrections at the order of the cooperon which restore the normalization of the probability. We study also the angular and temporal (diffusive wave spectroscopy) correlation functions of speckle patterns. More particularly, we obtain a closed expression of the contribution to the time correlation function which is equivalent to the universal conductance fluctuations. Finally, the notion of dephasing is discussed and implemented for the case of the dephasing induced by the internal Zeeman degrees of freedom of cold atomic gases.

Semiclassical spectrum of integrable systems in a magnetic field

In these notes we discuss the topological nature of some problems in condensed matter physics. This topic has been widely studied in various contexts. In statistical mechanics, the possible stable defects in an ordered system have been classified according to the nature of the order parameter (e.g. scalar, vector, matrix) and the space dimensionality of the system using homotopy groups [1]. Then, the discovery of the quantum Hall effects and the role played by stable integers or rational numbers for systems with few or no conserved quantum symmetries have motivated several topological models of quantum condensed matter systems [2,4]. A combination of these two ideas of defects classification and microscopic quantum models has been used in the description of superfluid 3He [5].