Exact Random Walk Distributions using Noncommutative Geometry

Using the results obtained by the non commutative geometry techniques applied to the Harper equation, we derive the areas distribution of random walks of length $N$ on a two-dimensional square lattice for large $N$, taking into account finite size contributions.Comment: Latex, 3 pages, 1 figure, to be published in J. Phys. A : Math. Ge

Two interacting Hofstadter butterflies

The problem of two interacting particles in a quasiperiodic potential is addressed. Using analytical and numerical methods, we explore the spectral properties and eigenstates structure from the weak to the strong interaction case. More precisely, a semiclassical approach based on non commutative geometry techniques permits to understand the intricate structure of such a spectrum. An interaction induced localization effect is furthermore emphasized. We discuss the application of our results on a two-dimensional model of two particles in a uniform magnetic field with on-site interaction.Comment: revtex, 12 pages, 11 figure

Quantum Computation of a Complex System : the Kicked Harper Model

The simulation of complex quantum systems on a quantum computer is studied, taking the kicked Harper model as an example. This well-studied system has a rich variety of dynamical behavior depending on parameters, displays interesting phenomena such as fractal spectra, mixed phase space, dynamical localization, anomalous diffusion, or partial delocalization, and can describe electrons in a magnetic field. Three different quantum algorithms are presented and analyzed, enabling to simulate efficiently the evolution operator of this system with different precision using different resources. Depending on the parameters chosen, the system is near-integrable, localized, or partially delocalized. In each case we identify transport or spectral quantities which can be obtained more efficiently on a quantum computer than on a classical one. In most cases, a polynomial gain compared to classical algorithms is obtained, which can be quadratic or less depending on the parameter regime. We also present the effects of static imperfections on the quantities selected, and show that depending on the regime of parameters, very different behaviors are observed. Some quantities can be obtained reliably with moderate levels of imperfection, whereas others are exponentially sensitive to imperfection strength. In particular, the imperfection threshold for delocalization becomes exponentially small in the partially delocalized regime. Our results show that interesting behavior can be observed with as little as 7-8 qubits, and can be reliably measured in presence of moderate levels of internal imperfections

Double butterfly spectrum for two interacting particles in the Harper model

We study the effect of interparticle interaction $U$ on the spectrum of the Harper model and show that it leads to a pure-point component arising from the multifractal spectrum of non interacting problem. Our numerical studies allow to understand the global structure of the spectrum. Analytical approach developed permits to understand the origin of localized states in the limit of strong interaction $U$ and fine spectral structure for small $U$.Comment: revtex, 4 pages, 5 figure

Exploiting the interplay between cross-sectional and longitudinal data in Class III malocclusion patients

The aim of the study was to investigate how to improve the forecasting of craniofacial unbalance risk during growth among patients affected by Class III malocclusion. To this purpose we used computational methodologies such as Transductive Learning (TL), Boosting (B), and Feature Engineering (FE) instead of the traditional statistical analysis based on Classification trees and logistic models. Such techniques have been applied to cephalometric data from 728 cross-sectional untreated Class III subjects (6–14 years of age) and from 91 untreated Class III subjects followed longitudinally during the growth process. A cephalometric analysis comprising 11 variables has also been performed. The subjects followed longitudinally were divided into two subgroups: favourable and unfavourable growth, in comparison with normal craniofacial growth. With respect to traditional statistical predictive analytics, TL increased the accuracy in identifying subjects at risk of unfavourable growth. TL algorithm was useful in diffusion of information from longitudinal to cross-sectional subjects. The accuracy in identifying high-risk subjects to growth worsening increased from 63% to 78%. Finally, a further increase in identification accuracy, up to 83%, was produced by FE. A ranking of important variables in identifying subjects at risk of growth worsening, therefore, has been obtained

Interaction induced delocalisation for two particles in a periodic potential

We consider two interacting particles evolving in a one-dimensional periodic structure embedded in a magnetic field. We show that the strong localization induced by the magnetic field for particular values of the flux per unit cell is destroyed as soon as the particles interact. We study the spectral and the dynamical aspects of this transition.Comment: 4 pages, 5 EPS figures, minor misprints correcte

The Flux-Phase of the Half-Filled Band

The conjecture is verified that the optimum, energy minimizing magnetic flux for a half-filled band of electrons hopping on a planar, bipartite graph is $\pi$ per square plaquette. We require {\it only} that the graph has periodicity in one direction and the result includes the hexagonal lattice (with flux 0 per hexagon) as a special case. The theorem goes beyond previous conjectures in several ways: (1) It does not assume, a-priori, that all plaquettes have the same flux (as in Hofstadter's model); (2) A Hubbard type on-site interaction of any sign, as well as certain longer range interactions, can be included; (3) The conclusion holds for positive temperature as well as the ground state; (4) The results hold in $D \geq 2$ dimensions if there is periodicity in $D-1$ directions (e.g., the cubic lattice has the lowest energy if there is flux $\pi$ in each square face).Comment: 9 pages, EHL14/Aug/9

Self-Organized Nanorod Arrays for Large-Area Surface-Enhanced Infrared Absorption

Capabilities of highly sensitive surface-enhanced infrared absorption (SEIRA) spectroscopy are demonstrated by exploiting large-area templates (cm2) based on self-organized (SO) nanorod antennas. We engineered highly dense arrays of gold nanorod antennas featuring polarization-sensitive localized plasmon resonances, tunable over a broadband near- and mid-infrared (IR) spectrum, in overlap with the so-called "functional group" window. We demonstrate polarization-sensitive SEIRA activity, homogeneous over macroscopic areas and stable in time, by exploiting prototype self-assembled monolayers of IR-active octadecanthiol (ODT) molecules. The strong coupling between the plasmonic excitation and molecular stretching modes gives rise to characteristic Fano resonances in SEIRA. The SO engineering of the active hotspots in the arrays allows us to achieve signal amplitude improved up to 5.7%. This figure is competitive to the response of lithographic nanoantennas and is stable when the optical excitation spot varies from the micro- to macroscale, thus enabling highly sensitive SEIRA spectroscopy with cost-effective nanosensor devices

Bloch Electrons in a Magnetic Field - Why Does Chaos Send Electrons the Hard Way?

We find that a 2D periodic potential with different modulation amplitudes in x- and y-direction and a perpendicular magnetic field may lead to a transition to electron transport along the direction of stronger modulation and to localization in the direction of weaker modulation. In the experimentally accessible regime we relate this new quantum transport phenomenon to avoided band crossing due to classical chaos.Comment: 4 pages, 3 figures, minor modifications, PRL to appea