Fractional-filling Mott domains in two dimensional optical superlattices

Ultracold bosons in optical superlattices are expected to exhibit fractional-filling insulating phases for sufficiently large repulsive interactions. On strictly 1D systems, the exact mapping between hard-core bosons and free spinless fermions shows that any periodic modulation in the lattice parameters causes the presence of fractional-filling insulator domains. Here, we focus on two recently proposed realistic 2D structures where such mapping does not hold, i.e. the two-leg ladder and the trimerized kagome' lattice. Based on a cell strong-coupling perturbation technique, we provide quantitatively satisfactory phase diagrams for these structures, and give estimates for the occurrence of the fractional-filling insulator domains in terms of the inter-cell/intra-cell hopping amplitude ratio.Comment: 4 pages, 3 figure

Topological regulation of activation barriers on fractal substrates

We study phase-ordering dynamics of a ferromagnetic system with a scalar order-parameter on fractal graphs. We propose a scaling approach, inspired by renormalization group ideas, where a crossover between distinct dynamical behaviors is induced by the presence of a length $\lambda$ associated to the topological properties of the graph. The transition between the early and the asymptotic stage is observed when the typical size $L(t)$ of the growing ordered domains reaches the crossover length $\lambda$. We consider two classes of inhomogeneous substrates, with different activated processes, where the effects of the free energy barriers can be analytically controlled during the evolution. On finitely ramified graphs the free energy barriers encountered by domains walls grow logarithmically with $L(t)$ while they increase as a power-law on all the other structures. This produces different asymptotic growth laws (power-laws vs logarithmic) and different dependence of the crossover length $\lambda$ on the model parameters. Our theoretical picture agrees very well with extensive numerical simulations.Comment: 13 pages, 4 figure

Superdiffusion and Transport in 2d-systems with L\'evy Like Quenched Disorder

We present an extensive analysis of transport properties in superdiffusive two dimensional quenched random media, obtained by packing disks with radii distributed according to a L\'evy law. We consider transport and scaling properties in samples packed with two different procedures, at fixed filling fraction and at self-similar packing, and we clarify the role of the two procedures in the superdiffusive effects. Using the behavior of the filling fraction in finite size systems as the main geometrical parameter, we define an effective L\'evy exponents that correctly estimate the finite size effects. The effective L\'evy exponent rules the dynamical scaling of the main transport properties and identify the region where superdiffusive effects can be detected.Comment: 12 pages, 19 figure

On a class of rational matrices and interpolating polynomials related to the discrete Laplace operator

Let \dlap be the discrete Laplace operator acting on functions (or rational matrices) $f:\mathbf{Q}_L\to\mathbb{Q}$, where $\mathbf{Q}_L$ is the two dimensional lattice of size $L$ embedded in $\mathbb{Z}_2$. Consider a rational $L\times L$ matrix $\mathcal{H}$, whose inner entries $\mathcal{H}_{ij}$ satisfy \dlap\mathcal{H}_{ij}=0. The matrix $\mathcal{H}$ is thus the classical finite difference five-points approximation of the Laplace operator in two variables. We give a constructive proof that $\mathcal{H}$ is the restriction to $\mathbf{Q}_L$ of a discrete harmonic polynomial in two variables for any $L>2$. This result proves a conjecture formulated in the context of deterministic fixed-energy sandpile models in statistical mechanics.Comment: 18 pag, submitted to "Note di Matematica

Anomalous transmission and drifts in one-dimensional Levy structures

We study the transmission of random walkers through a finite-size inhomogeneous material with a quenched, long-range correlated distribution of scatterers. We focus on a finite one-dimensional structure where walkers undergo random collisions with a subset of sites distributed on deterministic (Cantor-like) or random positions, with L\'evy spaced distances. Using scaling arguments, we consider stationary and time-dependent transmission and we provide predictions on the scaling behaviour of particle current as a function of the sample size. We show that, even in absence of bias, for each single realization a non-zero drift can be present, due to the intrinsic asymmetry of each specific arrangement of the scattering sites. For finite systems, this average drift is particularly important for characterizing the transmission properties of individual samples. The predictions are tested against the numerical solution of the associated master equation. A comparison of different boundary conditions is given.Comment: Submitted to Chaos, Solitons and Fractal

Fast rare events in exit times distributions of jump processes

Rare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We study three jump processes that are used to model a wide class of stochastic processes ranging from biology to transport in disordered systems, ecology and finance. We consider discrete time random-walks, continuous time random-walks and the L\'evy-Lorentz gas and determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits in a very short time at a large distance opposite to the starting point. For this estimation we use the so called big jump principle, and we show that in the regime of fast rare events the exit time distributions are not exponentially suppressed, even in the case of normal diffusion. This implies that fast rare events are actually much less rare than predicted by the usual estimates of large deviations and can occur on timescales orders of magnitude shorter than expected. Our results are confirmed by extensive numerical simulations.Comment: 6 pages, 4 figure

Phase ordering in disordered and inhomogeneous systems

We study numerically the coarsening dynamics of the Ising model on a regular lattice with random bonds and on deterministic fractal substrates. We propose a unifying interpretation of the phase-ordering processes based on two classes of dynamical behaviors characterized by different growth-laws of the ordered domains size - logarithmic or power-law respectively. It is conjectured that the interplay between these dynamical classes is regulated by the same topological feature which governs the presence or the absence of a finite-temperature phase-transition.Comment: 15 pages, 7 figures. To appear on Physical Review E (2015