Physical aspects (June 13 - 17)

Mind the boat

Departure from La Rochelle sunday (june 12) at 7pm, no boat on monday!
Departure from Boyardville saturday (june 18) at 1.15pm, no boat on friday!
See full timetables.

Schedule

Afternoons are devoted to talks and free discussions.You can already propose a talk by email.

Lectures

Anuradha Jagannathan: Quantum particles on tilings

In these lectures we will consider electrons and bosons in a quasiperiodic environment, and discuss some physical phenomena due to the highly organized spatial structure:

1. Experiments and , the nearly free electron model in quasicrystals, and phenomenology. Pseudogap, wavepackets, transport.
2. Tight-binding models on tilings. a) The Fibonacci chain spectrum and eigenstates. Anomalous diffusion. b) Models on two dimensional tilings. Electronic states, characteristic spectral features.
3. Magnetism in quasicrystals. Quantum spins, fluctuations and correlations.

Renaud Leplaideur: Freezing phase transitions with support in quasi-crystals Notes Slides

The goal of this mini-course is to present new developments in ergodic theory and thermodynamic formalism. Thermodynamic formalism has been introduced into the setting of dynamical systems and ergodic theory in the 70's by Sinai, Ruelle and Bowen. It usually deals with one-dimensional lattice, and dynamics is usually seen as the action of the time. The main point is that people in ergodic theory use vocabulary from Physics but it indeed turns out that the notions behind are sensibly different.

In the first lecture we will explain how the thermodynamic formalism from ergodic viewpoint is related to the Curry-Weiss viewpoint for Ising model or Potts model (also called mean-field interactions). We shall give a kind of dictionary that translates one notion in one setting into the other setting. The second and third lectures are devoted to recent works concerning freezing phase transitions. We will explain how we can construct examples with freezing phase transitions reaching a quasi-crystal state at positive temperature. We will first recall the definitions of the objects and introduce the main tool which is the Ruelle-Perron-Frobenius operator for inducing schemes. We will then prove that a freezing phase transition occurs. For that, we shall explore how orbits (in the shift) may approach the quasi-crystal.

Ron Lifshitz: Symmetry, excitations, defects, and thermodynamic stability of quasicrystals

The discovery of quasicrystals has changed our view of some of the most basic notions related to the condensed state of matter, leading to a new underlying paradigm for the notion of a crystal as a solid with long-range positional order. This paradigm shift brought about a new fundamental understanding of the basic characteristics of a crystal—its symmetry, its elementary collective excitations, and its topological defects. Before we knew about quasicrystals, it was believed that crystals break t he continuous translation and rotation symmetries of the liquid-phase into a discrete lattice of translations, and a finite point group of rotations. Quasicrystals, on the other hand, possess no such symmetries—there are no translations, nor, in general, are there any rotations, leaving them invariant. Does this imply that no symmetry is left, or that the meaning of symmetry in crystallography should be revised? Without an underlying lattice, can we even describe the symmetry-breaking transition that occurs upon solidification? Can we explain the thermodynamic stability of the solid phase that is formed and possibly control its self-assembly? Can we describe the elementary excitations? Can we talk about dislocations without a lattice? If so, how can we characterize them?

In my lectures I shall review these questions, related to the symmetry-breaking transition from a liquid to a crystal, using the notion of indistinguishability, which was introduced years ago by Rokhsar, Wright, and Mermin. This will allow me to provide some physical and geometric intuition, eventually leading to the introduction of some rather simple free-energy models that give rise to controlled self-assembly of thermodynamically stable quasicrystals.

Afternoon talks

• Erik Akkerman : Topology and fractals : measuring Chernnumbers with waves in quasi-crystals (slides)
• Jean-Noël Fuchs : Magnetic field properties of itinerant electrons in a quasicrystal (slides)
• Samuel Petite : On the Frenkel-Kontarova model associated to a quasicrystal
• Lorenzo Sadun : Tutorial on tiling dynamicsl systems (slides)
• Thomas Fernique : Matching rules for substitution tilings
• Pavel Kalouguine : Matching rules for cut & project tilings
• Mikhail Skopenkov : Tiling by rectangles and alternating current (slides)
• Johannes Kellendonk : Tutorial on K-theory and cohomology
• Siegfried Beckus : Continuity of spectra
• Pavel Kalouguine : An Ansatz for exact wavefunctions in quasicrystals I (slides)
• Nicolas Macé : An Ansatz for exact wavefunctions in quasicrystals II
• Thomas Fernique : Complete proof of the Mozes theorem (substitutive tilings are sofic)
• Eric Akkerman : Chern numbers II(slides)
• Frédéric Piéchon : Multifractal properties of the energy spectrum and wave functions on the Fibonacci chain (slides)
• Group discussion on the thermodynamic formalism in physics and mathematics
• Jordan Emme : Thermodynamic formalism and k-bonacci substitutions
• Nishant Chandgotia : Markov random fields and Gibbs states (slides)