Disorder and interference: localization phenomena

The specific problem we address in these lectures is the problem of transport and localization in disordered systems, when interference is present, as characteristic for waves, with a focus on realizations with ultracold atoms.Comment: Notes of a lecture delivered at the Les Houches School of Physics on "Ultracold gases and quantum information" 2009 in Singapore. v3: corrected mistakes, improved script for numerics, Chapter 9 in "Les Houches 2009 - Session XCI: Ultracold Gases and Quantum Information" edited by C. Miniatura et al. (Oxford University Press, 2011

Interaction instability of localization in quasiperiodic systems

Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.Comment: 10 pages; (v2: additional explanations, data, and references

Spatially heterogeneous dynamics in a thermosensitive soft suspension before and after the glass transition

The microscopic dynamics and aging of a soft thermosensitive suspension was investigated by looking at the thermal fluctuations of tracers in the suspension. Below and above the glass transition, the dense microgel particles suspension was found to develop an heterogeneous dynamics, featured by a non Gaussian Probability Distribution Function (PDF) of the probes' displacements, with an exponential tail. We show that non Gaussian shapes are a characteristic of the ensemble-averaged PDF, while local PDF remain Gaussian. This shows that the scenario behind the non Gaussian van Hove functions is a spatially heterogeneous dynamics, characterized by a spatial distribution of locally homogeneous dynamical environments through the sample, on the considered time scales. We characterize these statistical distributions of dynamical environments, in the liquid, supercooled, and glass states, and show that it can explain the observed exponential tail of the van Hove functions observed in the concentrated states. The intensity of spatial heterogeneities was found to amplify with increasing volume fraction. In the aging regime, it tends to increase as the glass gets more arrested.Comment: 19 pages, 10 figures, Soft Matter accepte

Theoretical study of scattering in graphene ribbons in the presence of structural and atomistic edge roughness

We investigate the diffusive electron-transport properties of charge-doped graphene ribbons and nanoribbons with imperfect edges. We consider different regimes of edge scattering, ranging from wide graphene ribbons with (partially) diffusive edge scattering to ribbons with large width variations and nanoribbons with atomistic edge roughness. For the latter, we introduce an approach based on pseudopotentials, allowing for an atomistic treatment of the band structure and the scattering potential, on the self-consistent solution of the Boltzmann transport equation within the relaxation-time approximation and taking into account the edge-roughness properties and statistics. The resulting resistivity depends strongly on the ribbon orientation, with zigzag (armchair) ribbons showing the smallest (largest) resistivity and intermediate ribbon orientations exhibiting intermediate resistivity values. The results also show clear resistivity peaks, corresponding to peaks in the density of states due to the confinement-induced subband quantization, except for armchair-edge ribbons that show a very strong width dependence because of their claromatic behavior. Furthermore, we identify a strong interplay between the relative position of the two valleys of graphene along the transport direction, the correlation profile of the atomistic edge roughness, and the chiral valley modes, leading to a peculiar strongly suppressed resistivity regime, most pronounced for the zigzag orientation.Comment: 13 pages, 7 figure

Microscopic Aspects of Stretched Exponential Relaxation (SER) in Homogeneous Molecular and Network Glasses and Polymers

Because the theory of SER is still a work in progress, the phenomenon itself can be said to be the oldest unsolved problem in science, as it started with Kohlrausch in 1847. Many electrical and optical phenomena exhibit SER with probe relaxation I(t) ~ exp[-(t/{\tau}){\beta}], with 0 < {\beta} < 1. Here {\tau} is a material-sensitive parameter, useful for discussing chemical trends. The "shape" parameter {\beta} is dimensionless and plays the role of a non-equilibrium scaling exponent; its value, especially in glasses, is both practically useful and theoretically significant. The mathematical complexity of SER is such that rigorous derivations of this peculiar function were not achieved until the 1970's. The focus of much of the 1970's pioneering work was spatial relaxation of electronic charge, but SER is a universal phenomenon, and today atomic and molecular relaxation of glasses and deeply supercooled liquids provide the most reliable data. As the data base grew, the need for a quantitative theory increased; this need was finally met by the diffusion-to-traps topological model, which yields a remarkably simple expression for the shape parameter {\beta}, given by d*/(d* + 2). At first sight this expression appears to be identical to d/(d + 2), where d is the actual spatial dimensionality, as originally derived. The original model, however, failed to explain much of the data base. Here the theme of earlier reviews, based on the observation that in the presence of short-range forces only d* = d = 3 is the actual spatial dimensionality, while for mixed short- and long-range forces, d* = fd = d/2, is applied to four new spectacular examples, where it turns out that SER is useful not only for purposes of quality control, but also for defining what is meant by a glass in novel contexts. (Please see full abstract in main text

Obtaining localization properties efficiently using the Kubo-Greenwood formalism

We establish, through numerical calculations and comparisons with a recursive Green's function based implementation of the Landauer-B\"uttiker formalism, an efficient method for studying Anderson localization in quasi-one-dimensional and two-dimensional systems using the Kubo-Greenwood formalism. Although the recursive Green's function method can be used to obtain the localization length of a mesoscopic conductor, it is numerically very expensive for systems that contain a large number of atoms transverse to the transport direction. On the other hand, linear-scaling has been achieved with the Kubo-Greenwood method, enabling the study of effectively two-dimensional systems. While the propagating length of the charge carriers will eventually saturate to a finite value in the localized regime, the conductances given by the Kubo-Greenwood method and the recursive Green's function method agree before the saturation. The converged value of the propagating length is found to be directly proportional to the localization length obtained from the exponential decay of the conductance.Comment: 7 pages, 6 figure

Non-Markovian data-driven modeling of single-cell motility

Trajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by thenon-Markovian version of the Langevin equation that includes an arbitrary memory function. When averagedover cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterizedby Gaussian velocity distributions. Accordingly, the data are described by a linear memory model whichincludes different random walk models that were previously used to account for various aspects of cell motilitysuch as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memoryfunction is extracted from the trajectory data without restrictions or assumptions, thus making our approachtruly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayedsingle-exponential negative friction, which clearly distinguishes cell motion from the simple persistent randomwalk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extractedmemory function we formulate a generalized exactly solvable cell migration model which indicates thatnegative friction generates cell persistence over long timescales. The nonequilibrium character of cell motionis investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model thatinvolves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process