DISORDER

We present a method that allows us to find the asymptotic form of various characteristics of disordered systems in the strong localization regime, i.e., when either the random potential is big or the energy is close to a spectral edge. The method is based on the hypothesis that the relevant realizations of the random potential in the strong localization regime have the form of a collection of deep random wells that are uniformly and chaotically distributed in space with a sufficiently small density. Assuming this and using the density expansion, we show first that the density of wells coincides in leading order with the density of states. Thus the density of states is in fact the small parameter of the theory in the strong localization regime. Then we derive the Mott formula for the low frequency conductivity and the asymptotic formulae for certain two-point correlators when the difference of the respective energies is small.

The Anderson localization problem in one and two dimensions is solved analytically via the calculation of the generalized Lyapunov exponents. This is achieved by making use of signal theory. The phase diagram can be analysed in this way. In the one-dimensional case all states are localized for arbitrarily small disorder in agreement with existing theories. In the two-dimensional case for larger energies and large disorder all states are localized but for certain energies and small disorder extended and localized states coexist. The phase of delocalized states is marginally stable. We demonstrate that the metal–insulator transition should be interpreted as a first-order phase transition. Consequences for perturbation approaches, the problem of self-averaging quantities and numerical scaling are discussed.

We calculate two-point energy level correlation function in weakly disordered metallic grain by taking account of localization corrections to the universal random matrix result. Using supersymmetric nonlinear σ model and exactly integrating spatially homogeneous modes, we derive the expression valid for arbitrary energy differences from quantum to diffusive regime for the system with broken time reversal symmetry. Our result coincides with that obtained by Andreev and Altshuler (1995 Phys. Rev. Lett . 72 902) where homogeneous modes are perturbatively treated.

In this talk we review our theoretical understanding of spin glasses paying a particular attention to the basic physical ideas. We introduce the replica method and we describe its probabilistic consequences (we stress the recently discovered importance of stochastic stability). We show that the replica method is not restricted to systems with quenched disorder. We present the consequences on the dynamics of the system when it slows approaches equilibrium are presented: they are confirmed by large scale simulations, while we are still awaiting for a direct experimental verification.

In these two lectures I review our theoretical understanding of spin glasses paying a particular attention to the basic physical ideas. We introduce the replica method and we describe its probabilistic consequences (we stress the recently discovered importance of stochastic stability). We show that the replica method is not restricted to systems with quenched disorder. We present the consequences on the dynamics of the system when it slows approaches equilibrium are presented: they are confirmed by large scale simulations, while we are still awaiting for a direct experimental

The effective action for the current and density is shown to satisfy an evolution equation, constructed by the analogy of the functional renormalization group, which describes the dependence of the one-particle irreducible vertex functions on the strength of the disorder and the Coulomb interaction. No small parameter is assumed in deriving the evolution equation. The case of the non-interacting electron gas is discussed in more details. The effective action is constructed in the initial condition for weakly fluctuating impurity field in the framework of the gradient expansion. It is conjectured that non-linearities should drive the transport coefficients to zero at finite strength of the disorder in a manner analogous to a formal diffusion process and the quartic order terms of the gradient expansion would be the key to the localized phase.

The functional renormalization group method is used to take into account the vacuum polarization around localized bound states generated by external potential. The application to Atomic Physics leads to improved Hartree-Fock and Kohn-Sham equations in a systematic manner within the framework of the Density Functional Theory. Another application to Condensed Matter Physics consists of an algorithm to compute quenched averages with or without Coulomb interaction in a non-perturbative manner.

In the present Ph.D. dissertation I present results concerning disordered and frustrated models of relevance in statistical mechanics and in combinatorial optimization. As an application of spin glass theory I study the disordered and frustrated Blume-Emery-Griffiths model. The model is treated in its mean-field approximation using replicas. Within the Ansatz of replica-symmetry, I present a complete numerical solution; I also discuss effects of replica symmetry breaking. The stability of the RS solution is studied and the regions of instability inferred. The phase diagram exhibits first and second order transitions. The tricritical point is still present in the frustrated model, in agreement with former work. A version of the BEG model with disordered chemical potential is also studied. The calculations confirm that the disorder decreases the tricritical temperature. Next, I consider the graph partitioning problem, a combinatorial optimization problem, which, from the point of view of statistical mechanics is a spin glass model with the constraint of zero magnetisation. I focus on the statistical properties of low energy solutions generated by heuristic algorithms designed to solve such hard combinatorial optimization problems. Several heuristics proposed to solve this problem were implemented. Scaling laws are obtained; in particular, the average cost and its variance grow linearly with the number of vertices of the graphs. As a consequence the cost found by the heuristics is self-averaging. I suggest that this property is quite general, valid for random solutions, quasi-optimal solutions, and probably for the optimum solutions, too. Furthermore a ranking method is proposed and illustrated on an ensemble of graph partitioning problems. This ranking procedure takes into account the quality of the solution as well as the time necessary to find that solution. In the third part of this dissertation I study in detail the zero-temperatures properties of spin glasses on sparse random graphs with fixed connectivity. Spin glasses on these graphs may be considered as a more realistic approximation to real spin glasses as represented by the model of Sherrington and Kirkpatrick. I have designed a new algorithm for finding low energy states. Second, I present a method for deriving the ground state energy from heuristic algorithms, even though they are not guaranteed to find the optimum. Third, I present a numerical test of a conjecture due to Banavar, Sherrington and Sourlas, giving the large volume energy density of the ground states as function of the connectivity. The distribution of the order parameter is found to be non-trivial, and I give evidence for the presence of ultrametricity for all values of the connectivity. These results confirm the expectation that the remarquable properties of the infinite range Sherrington-Kirkpatrick model carry over to more realistic models, as for example the spin glass model on random graphs with finite connectivity studied in the present work.

Recently time-reversal techniques have emerged as a new, important and fascinating discipline within wave propagation. Many of the problems involved can best be understood, analysed and optimized based on a random field model for the medium. Here we discuss stable refocusing of second-order time-reversed reflections. This phenomenon may appear as surprising at first. However, we show how it can be understood in very simple terms viewing the wavefield as a stochastic process. We give sufficient conditions on Green's function of the propagation problem for the phenomenon to happen. In particular we discuss acoustic wave propagation in the regime of weak random medium fluctuations and explicitly give the derivation of stable refocusing in this case, illustrating it with numerical examples.