Bounds on the volume entropy and simplicial volume in Ricci curvature LpL^p bounded from below

Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $\pi:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of f\o\pi on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number and presentations of the fundamental group) in Ricci curvature $L^p$-bounded from below

Disorder effect on the localization/delocalization in incommensurate potential

The interplay between incommensurate (IC) and random potentials is studied in a two-dimensional symplectic model with the focus on localization/delocalization problem. With the IC potential only, there appear wavefunctions localized along the IC wavevector while extended perpendicular to it. Once the disorder potential is introduced, these turn into two-dimensional anisotropic metallic states beyond the scale of the elastic mean free path, and eventually becomes localized in both directions at a critical strength of the disorder. Implications of these results to the experimental observation of the IC-induced localization is discussed.Comment: 4 pages, 3 figures (7 files), RevTe

Staggered and extreme localization of electron states in fractal space

We present exact analytical results revealing the existence of a countable infinity of unusual single particle states, which are localized with a multitude of localization lengths in a Vicsek fractal network with diamond shaped loops as the 'unit cells'. The family of localized states form clusters of increasing size, much in the sense of Aharonov-Bohm cages [J. Vidal et al., Phys. Rev. Lett. 81, 5888 (1998)], but now without a magnetic field. The length scale at which the localization effect for each of these states sets in can be uniquely predicted following a well defined prescription developed within the framework of real space renormalization group. The scheme allows an exact evaluation of the energy eigenvalue for every such state which is ensured to remain in the spectrum of the system even in the thermodynamic limit. In addition, we discuss the existence of a perfectly conducting state at the band center of this geometry and the influence of a uniform magnetic field threading each elementary plaquette of the lattice on its spectral properties. Of particular interest is the case of extreme localization of single particle states when the magnetic flux equals half the fundamental flux quantum.Comment: 9 pages, 8 figure

An analytical law for size effects on thermal conductivity of nanostructures

The thermal conductivity of a nanostructure is sensitive to its dimensions. A simple analytical scaling law that predicts how conductivity changes with the dimensions of the structure, however, has not been developed. The lack of such a law is a hurdle in "phonon engineering" of many important applications. Here, we report an analytical scaling law for thermal conductivity of nanostructures as a function of their dimensions. We have verified the law using very large molecular dynamics simulations

Lubricated friction between incommensurate substrates

This paper is part of a study of the frictional dynamics of a confined solid lubricant film - modelled as a one-dimensional chain of interacting particles confined between two ideally incommensurate substrates, one of which is driven relative to the other through an attached spring moving at constant velocity. This model system is characterized by three inherent length scales; depending on the precise choice of incommensurability among them it displays a strikingly different tribological behavior. Contrary to two length-scale systems such as the standard Frenkel-Kontorova (FK) model, for large chain stiffness one finds that here the most favorable (lowest friction) sliding regime is achieved by chain-substrate incommensurabilities belonging to the class of non-quadratic irrational numbers (e.g., the spiral mean). The well-known golden mean (quadratic) incommensurability which slides best in the standard FK model shows instead higher kinetic-friction values. The underlying reason lies in the pinning properties of the lattice of solitons formed by the chain with the substrate having the closest periodicity, with the other slider.Comment: 14 pagine latex - elsart, including 4 figures, submitted to Tribology Internationa

Asymptotic energy profile of a wavepacket in disordered chains

We investigate the long time behavior of a wavepacket initially localized at a single site $n_0$ in translationally invariant harmonic and anharmonic chains with random interactions. In the harmonic case, the energy profile $\bar{< e_n(t)>}$ averaged on time and disorder decays for large $|n-n_0|$ as a power law $\bar{}\approx C|n-n_0|^{-\eta}$ where $\eta=5/2$ and 3/2 for initial displacement and momentum excitations, respectively. The prefactor $C$ depends on the probability distribution of the harmonic coupling constants and diverges in the limit of weak disorder. As a consequence, the moments $< m_{\nu}(t)>$ of the energy distribution averaged with respect to disorder diverge in time as $t^{\beta(\nu)}$ for $\nu \geq 2$, where $\beta=\nu+1-\eta$ for $\nu>\eta-1$. Molecular dynamics simulations yield good agreement with these theoretical predictions. Therefore, in this system, the second moment of the wavepacket diverges as a function of time despite the wavepacket is not spreading. Thus, this only criteria often considered earlier as proving the spreading of a wave packet, cannot be considered as sufficient in any model. The anharmonic case is investigated numerically. It is found for intermediate disorder, that the tail of the energy profile becomes very close to those of the harmonic case. For weak and strong disorder, our results suggest that the crossover to the harmonic behavior occurs at much larger $|n-n_0|$ and larger time.Comment: To appear in Phys. Rev.

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

One-dimensional quasiperiodic systems, such as the Harper model and the Fibonacci quasicrystal, have long been the focus of extensive theoretical and experimental research. Recently, the Harper model was found to be topologically nontrivial. Here, we derive a general model that embodies a continuous deformation between these seemingly unrelated models. We show that this deformation does not close any bulk gaps, and thus prove that these models are in fact topologically equivalent. Remarkably, they are equivalent regardless of whether the quasiperiodicity appears as an on-site or hopping modulation. This proves that these different models share the same boundary phenomena and explains past measurements. We generalize this equivalence to any Fibonacci-like quasicrystal, i.e., a cut and project in any irrational angle.Comment: 7 pages, 2 figures, minor change

Localization problem of the quasiperiodic system with the spin orbit interaction

We study one dimensional quasiperiodic system obtained from the tight-binding model on the square lattice in a uniform magnetic field with the spin orbit interaction. The phase diagram with respect to the Harper coupling and the Rashba coupling are proposed from a number of numerical studies including a multifractal analysis. There are four phases, I, II, III, and IV in this order from weak to strong Harper coupling. In the weak coupling phase I all the wave functions are extended, in the intermediate coupling phases II and III mobility edges exist, and accordingly both localized and extended wave functions exist, and in the strong Harper coupling phase IV all the wave functions are localized. Phase I and Phase IV are related by the duality, and phases II and III are related by the duality, as well. A localized wave function is related to an extended wave function by the duality, and vice versa. The boundary between phases II and III is the self-dual line on which all the wave functions are critical. In the present model the duality does not lead to pure spectra in contrast to the case of Harper equation.Comment: 10 pages, 11 figure