Localization of the eigenfunctions of quantum particles in a random potential was discovered by P.W. Anderson more than 50 years ago. In spite of its respectable maturity and intensive theoretical and experimental studies this field is far from being exhausted. Anderson localization was originally discovered in connection with spin relaxation and charge transport in disordered conductors. Later this phenomenon was observed for light, microwaves, sound, and more recently for cold atoms. Moreover, it became clear that the domain of applicability of the concept of localization is much broader. For example, it provides an adequate framework for discussing the transition between integrable and chaotic behavior in quantum systems. We will discuss the connection between the classical (Kolmogorov, Arnold, Moser theory) and quantum (Anderson Localization) transitions between integrable and chaotic types of behavior of complex physical systems. We will consider several examples of the manifestation of this paradigm - from adiabatic quantum computation to many-body statistical mechanics.

Interacting electrons on a square lattice, as described for instance by the Hubbard Hamiltonian, represent undoubtedly the central issue in the quest for a microscopic understanding of cuprate superconductors. This talk will present variational results for the 2D Hubbard model [1], which exhibit a superconducting ground state for a certain doping range, in surprisingly good agreement with experiments. Our ansatz belongs to a class of trial states which are "adiabatically" linked to a simple reference state and therefore not ideally suited for describing quantum phase transitions [2]. Possible ways out of this dilemma are briefly discussed.

[1] D. Baeriswyl, D. Eichenberger and M. Menteshashvili, New J. Phys. 11, 075010 (2009).
[2] D. Baeriswyl, to appear in Ann. Phys. (Berlin) 19 (2011).

I am going to review some of the topics addressed in the course of my scientific collaboration with Mario, focusing on connections between discretized (2 and 3d) models and topological quantum computation.

Two objects we classify together cannot be identical as we have to distinguish between them. Thus they can only be almost equal: every symmetry is -by itself- broken. Broken symmetries were, up to few decades ago, realized starting from exact mathematical symmetries and introducing the breaking at the physical level or in the states (as in spontaneous symmetry breaking) or in the operators (as in mass formulas for hadrons). Recently the possibility has appeared to put the breaking directly in the mathematics. This talk is devoted to discuss this new approach where deformations of Lie algebras, called quantum algebras, are introduced. Quantum algebras considered as Hopf algebras are not deformations of Lie algebras but deformations of Lie universal enveloping algebras, i.e. they are infinite dimensional deformation of the infinite dimensional algebra of the polynomials constructed on the Lie algebras generators. To close the play we have thus to mimic the Cartan work to individuate the quantum algebra canonical basis to be put in one-to-one correspondence with the Cartan basis of the Lie algebra and successively with the physical operators of the unbroken symmetry. Starting from the Lie bialgebra and using coassociativity (in physics, the composed system rule) and analyticity, a self-consistent perturbative approach allows us to obtain the corresponding quantum algebra in its canonical basis. A consistent extension to quantum superalgebras as well as to quantum Poisson algebras has been also developed.

TBA

Many systems approach equilibrium very slowly: the equilibration time becomes macroscopic and sometimes it is so large that it cannot be measured. Strong progresses have recently done in understanding the collective phenomena that are at the basis of their behavior.
This talk will contain:
a) A mini introduction to structural glasses and spin glasses.
b) A theoretical framework for interpreting these phenomena.
c) Aging in structural glasses and spin glasses (theoretical predictions, experimental and numerical results).
d) Generalized fluctuation dissipation relations and the definition of a scale dependent temperature.

Atomic Bose-Einstein condensates (BECs) trapped in optical lattices (OLs) have been the subject of great recent experimental and theoretical interest, both in their own right and as analog models of certain solid state systems. Recent studies of the leakage of a BEC trapped in an OL have shown that highly localized nonlinear excitations known as "Intrinsic Localized Modes" (ILMs) can prevent atoms from reaching the leaking boundaries, thereby slowing the decay of the condensate.
In this talk I report the results of a recent study¹ (conducted with Holger Hennig and Jerome Dorignac) of this problem. To understand the mechanism by which these ILMs enhance the trapping, we study the case of atom transport-"tunneling"-through an ILM on a nonlinear trimer. We show that this transport is related to the destabilization and subsequent motion of DB and that there exists a threshold in the total energy on the trimer that controls this destabilization. We find that this threshold and the resultant tunneling can be described analytically by defining a two-dimensional "Peierls-Nabarro" energy landscape which restricts the dynamics of the trimer to a limited region of phase space. We further establish that the value of the energy threshold is related to the Peierls- Nabarro barrier of a single ILM. We then embed our nonlinear trimer in an extended lattice and show numerically that the same destabilization mechanism applies in the extended lattice. Our results suggest a possible means for controlling the transmission of coherent atomic beams in interferometry and other processes.

¹ Phys. Rev. A 82, 053604 (2010)

A basic question of the Statistical Mechanics, starting from the Boltzmann's ergodic hypothesis, is the connection between the dynamics and the statistical properties. This is a rather difficult task and, in spite of the mathematical progress, for long time basically ergodic theory had a marginal relevance in the development of the statistical mechanics (at least in the physics community). Partially this was due to a widely spreaded opinion (based also on the belief of influential scientists as Landau) on the key role of the many degrees of freedom and the practical irrelevance of ergodicity. This point of view found a mathematical support on some results by Khinchin.
On the other hand the discovery of the deterministic chaos beyond its e undoubted relevance for many natural phenomena, showed how the similar statistical features observed in systems with many degrees of freedom, can be generated also by the presence of deterministic chaos in simple systems.
Even after many years, there is not a consensus on the basic conditions which should ensure the validity of the statistical mechanics. Roughly speaking the two extreme positions are the "traditional" one, for which the main ingredient is the presence of many degrees of freedom and the "innovative" one which considers chaos a crucial requirement to develop a statistical approach.
We discusse the role of ergodicity and chaos for the validity of statistical laws. Detailed studies show in a clear way that chaos is not a fundamental ingredient for the validity of the equilibrium statistical mechanics. Therefore the point of view that good statistical properties need chaos is unnecessarily demanding: even in the absence of chaos, one can have (according to Khichin ideas) a good agreement between the time averages and the predictions by the equilibrium statistical mechanics.
Basically one can say that thermodynamics can be seen as an emergent property of large systems.

We describe how negative temperature states may appear in the form of breathers coupled to a background in the Discrete Nonlinear Schrodinger Equation. We shall also discuss how transport properties are affected by the presence of such a kind of nonlinear excitations.