# Workshop Programme

## for period 18 - 22 August 2008

### Anderson Localization and Related Phenomena

18 - 22 August 2008

Timetable

 Monday 18 August 09:00-09:55 Registration 09:55-10:00 David Wallace - welcome 10:00-11:00 Thouless, D (Washington) Localization, conduction, and superfluidity Sem 1 Localization of electrons characterizes the difference between an electrical insulator and a metallic conductor at low temperatures, and localization of atoms is one of the characteristics of a solid as opposed to a fluid. I discuss the suggestion made forty years ago that a quantum solid might be a supersolid with superfluid properties, and the recent experiments that show indications of superfluid properties in some solid helium samples. The behavior of the `center of mass' of a system with periodic boundary conditions in one direction sheds light on the possibility of reduced moment of inertia in a solid. It seems unlikely that supersolidity is a property of a good crystal, but it has been argued that networks of dislocations or of grain boundaries could support a superfluid within real crystals. 11:00-11:30 Coffee 11:30-12:30 Kotani, S (Osaka) On distribution of zeros of polynomials approximating exponential type entire functions Sem 1 We discuss the limiting behavior of the distribution of zeros of a certain class of polynomials when their degrees increase to infinity. A special case is polynomials which are obtained as a Maclaulin expansion of the exponential function and in this case,after a suitable normalization, the set of zeros converges to a smooth curve with a singulality at 1. Several generalizations will be presented. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Gornyi, I (Forschungszentrum Karlsruhe) Anderson localization and electron-electron interaction Sem 1 I will discuss transport of interacting electrons in a low-dimensional disordered system at low temperature. In view of localization by disorder, the conductivity may only be non-zero due to electron-electron scattering. In this talk, I will review recent theoretical results on Anderson localization of interacting electrons. In particular, I will talk about Anderson localization and dephasing in strongly-correlated one-dimensional electron systems (disordered Luttinger liquids). The results I will describe were obtained in: I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, Phys. Rev. Lett. 95, 046404 (2005); Phys. Rev. Lett. 95, 206603 (2005); Phys. Rev. B 75, 085421 (2007); D.A. Bagrets, I.V. Gornyi, A.D. Mirlin, D.G. Polyakov, arXiv:0804.4887; A.G. Yashenkin, I.V. Gornyi, A.D. Mirlin, D.G. Polyakov, arXiv:0808.1093. 15:00-15:30 Tea 15:30-16:30 Yau, HT (Harvard) Local semicircle law and complete delocalization for Wigner random matrices Sem 1 We consider $N\times N$ Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{-1} (\log N)^8$. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the $\ell^\infty$-norm of the corresponding eigenvectors is of order $O(N^{-1/2})$, modulo logarithmic corrections. The upper bound $O(N^{-1/2})$ implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. 16:30-17:30 Wegner, F (Heidelberg) Cell model for scaling near the mobility edge Sem 1 On the basis of real-space renormalization (Z. Physik B25 (1976) 327) descriptive arguments for scaling at the mobility edge are given. Although these arguments are rather old, most of them are in qualitative agreement with today's knowledge on the behaviour around the mobility edge (non-linear sigma-model and renormalization). 17:30-18:30 Welcome Wine Reception 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Tuesday 19 August 09:00-10:00 Klein, A (UC, Irvine) Poisson statistics for eigenvalues of continuum random schr\"odinger operators Sem 1 We prove Poisson statistics for eigenvalues of random Schrödinger operators in the continuum. More specifically, we prove a Minami estimate for continuum Anderson Hamiltonians in the continuum and derive Poisson statistics for the eigenvalues in the localization region at the bottom of the spectrum. We also prove simplicity of the eigenvalues in that region. (Joint work with J.-M. Combes and F.Germinet) 10:00-11:00 Chalker, J (Oxford) Quantum and classical localisation transitions Sem 1 Localisation of a particle moving in a random environment may occur both quantum mechanically and with classical dynamics, but the phenomenon is very different in the two cases. I will discuss a class of quantum-mchanical localisation problems for which some physical quantities can be expressed exactly in terms of averages taken in a classical counterpart. The equivalence holds despite the fact that interference effects dominate the behaviour of the quantum systems. The models are network models belonging to class C in Zirnbauer's classification. The equivalence was first discovered in the context of the spin quantum Hall effect by Gruzberg, Ludwig and Read [Phys. Rev. Lett. 82, 4524 (1999)], and the results I will describe were obtained in collaboration with Beamond and Cardy [Phys Rev B 65, 214301 (2002) and with Ortuno and Somoza [in preparation]. 11:00-11:30 Coffee 11:30-12:30 Davies, B (King's College London) Spectral bounds for non-self-adjoint Anderson models Sem 1 This is a review of recent work by E B Davies, C Martinez and J Hinchcliffe on the spectral properties of some non-self-adjoint operators acting in Z^d. The operators include the one-dimensional NSA Anderson model of Hatano and Nelson. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Mirlin, A (Karlsruhe) Electron transport in disordered graphene Sem 1 Charge carriers in graphene have a relativistic (massless Dirac) spectrum, which is responsible for remarkable electronic properties of this 2D material. In this talk, I will first give a brief overview of the experimental results on transport properties of graphene samples. I will then present recent theoretical advances on electron transport in disordered graphene. Transport properties of the system are found to depend crucially on the character of disorder which influences the symmetry and the topology of the underlying field theory of disordered Dirac fermions. A particularly interesting situation emerges for long-range disorder which does not mix two Dirac "valleys" of the spectrum. Topics to be discussed: (i) Anderson localization or its absence; (ii) anomalous quantum Hall effect; (iii) evolution from the ballistic to the diffusive regime. The talk is based on results obtained in P.M. Ostrovsky, I.V. Gornyi, and A.D. Mirlin, Phys.Rev.B 74, 235443 (2006); Phys.Rev.Lett. 98, 256801 (2007); Eur.Phys.J.Special Topics 148, 63 (2007); Phys.Rev.B 77, 195430 (2008); A. Schuessler, P.M. Ostrovsky, I.V. Gornyi, and A.D. Mirlin, in preparation. 15:00-15:30 Tea 15:30-16:30 Goldsheid, I (QMUL) Non-Hermitian Anderson model on a strip: properties of eigenvalues and eigenfunctions Sem 1 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Wednesday 20 August 09:00-10:00 Gershenson, M (Rutgers, New Jersey) Interaction effects in the conductivity of high-mobility Si MOSFETs Sem 1 The beginning of the 80s witnessed a triumph of the scaling theory of localization and the theory of electron-electron interactions in disordered conductors: the low-temperature behavior of the conductivity of numerous low-dimensional systems has been successfully attributed to the quantum interference effects. Curiously, application of these ideas to the two-dimensional electron liquid in Si MOSFETs - one of the most ubiquitous 2D systems - remained a challenge for more than 25 years. Over the last decade, an increase of the conductivity with cooling was observed for many high-mobility and low-density systems (p- and n-type GaAs/AlGaAs, SiGe, etc.), and the terms "anomalous metal in 2D" and "apparent 2D metal-insulator transition" have been introduced. I will discuss recent progress in our understanding of these phenomena, in particular, the "metallicity" of high-mobility Si MOSFETs and the renormalization of Fermi-liquid parameters at low densities. 10:00-11:00 Graf, GM (ETH Z\"urich) A variant of an estimate by Minami Sem 1 In the context of the Anderson model, Minami proved a Wegner type bound on the expectation of 2 x 2 determinant of Green's functions. We generalize it so as to allow for a magnetic field, as well as to determinants of higher order (joint work with A. Vaghi). Related recent work by others will be reviewed. 11:00-11:30 Coffee 12:30-13:30 Lunch at Wolfson Court 19:00-23:00 Garden Party at St Edmund's College
 Thursday 21 August 09:00-10:00 Efetov, K (Ruhr-Universit\"at Bochum) Anderson localization in 2D systems : from Schrodinger to Dirac stochastic equations Sem 1 10:00-11:00 Disertori, M (Rouen) A supersymmetric model for quantum diffusion in 3d Sem 1 Supersymmetric approach has proved to be a powerful tool for the study of random systems in mesoscopic physics, where classical techniques do not apply. I'll give an introduction to the technique in the case of a hyperbolic sigma model. This model is expected to describe qualitatively properties of random band matrices in d dimensions such as localization and delocalization. (Joint work with T. Spencer and M. Zirnbauer) 11:00-11:30 Coffee 11:30-12:30 Muller-Hill, J (Cologne) Hyperbolic Hubbard-Stratonovich transformations Sem 1 An important application for hyperbolic Hubbard-Stranonovich transformations is the derivation of hyperbolic sigma models. We will explain the Schaefer-Wegner and Pruisken-Schaefer variants of the transformation. Until recently the mathematical status of the latter was unclear. We will give a proof of the Pruisken-Schaefer variant of the transformation, and clarify the connection to the Schaefer-Wegner variant 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Schenker, J (Michigan State) Diffusion of wave packets in a Markov random potential Sem 1 We show the square amplitude of a tight binding wave packet propagating in a sufficiently weak time dependent potential which evolves according to a stationary Markov proces converges, after diffusive rescaling, to a solution of a heat equation. 15:00-15:30 Tea 15:30-16:30 Schulz-Baldes, H (Erlangen-Nurnberg) The random phase hypothesis for quasi-1D random media Sem 1 The random transfer matrices of a quasi-1D system naturally act on the Grassmannian of symplectic frames, which is a compact symmetric space. The random phase hypothesis (RPH) claims that the invariant distribution is given by the Haar measure on the elliptic (or open) channels. This allows to derive the DKMP equations which in turn have a large number of applications. A general criterion is presented allowing to verify a weak form of the RPH in a perturbative situation. The criterion can be verified for the Wegner N-orbital model and this allows to prove that its Lyapunov spectrum is equidistant. Other applications concern the localization length of the Anderson model on a strip (giving a generalization of the 1d-Thouless formula) as well as a perturbative formula for the density of states. 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Friday 22 August 09:00-10:00 Pastur, L (Kharkov) On the Mott formula for the low frequency conductivity: a review Sem 1 10:00-11:00 Fishman, S (Technion) Anderson localisation for the nonlinear Schroedinger Equation (NLSE): results and puzzles Sem 1 The NLSE is relevant for the explorations of Bose-Einstein Condensates and for Nonlinear Classical Optics. A natural question is whether Anderson Localization survives the effect of nonlinearities in one dimension. Relevant experimental, numerical, heuristic and rigorous results will be presented. A perturbation expansion in the nonlinear term was developed and used to obtain a rigorous bound on the spreading for short times. In particular it is found that exponential localization holds at least for time scales inversely proportional to the square of the nonlinearity. Conjectures on the long time behavior will be presented. The work reported in the talk was done in collaboration with Avy Soffer and Yevgeny Krivolapov. 11:00-11:30 Coffee 11:30-12:30 Garcia-Garcia, A (Princeton) Anderson localization: from theoretical aspects to applications in QCD and cold atoms Sem 1 I summarize some of my recent research on Anderson localization and the metal-insulator transition. In the first part I discuss theoretical aspects: 1. I combine a selfconsistent treatment with the one parameter scaling theory to provide explicit expressions of typical quantities characterizing the metal insulator transition in d > 2 such as the critical exponent which controls the divergence of the localization length. 2. I explore under what circumstances a band of metallic states exists in 1d systems with correlated disorder. I relate the absence of localization with a certain degree of differentiability of the random potential. The second part is devoted to applications: 1. I adapt the one parameter scaling theory to quantum chaos in order to determine in what situations Anderson localization is expected in chaotic but not strictly random potentials. Potential applications to experimental tests of localization in cold atoms physics are briefly discussed. 2. I introduce Anderson localization in the context of Quantum Chromodynamics (QCD). Then I show that at the same time that the chiral phase transition occurs the spectrum of the QCD Dirac operator undergoes a metal-insulator transition. Finally I give a list of challenging problems for the future. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Oganesyan, V (Yale) Quantum vs. classical localization of excited many-particle states Sem 1 I shall discusss some recent ideas and numerical studies of localization of interacting particles with a finite energy per particle. 15:00-15:30 Tea 18:45-19:30 Dinner at Wolfson Court (Residents Only)