Superdiffusion and Transport in 2d-systems with Levy 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 Levy 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 Levy exponents that correctly estimate the finite size effects. The effective Levy exponent rules the dynamical scaling of the main transport properties and identify the region where superdiffusive effects can be detected.

Scattering lengths and universality in superdiffusive Levy materials

We study the effects of scattering lengths on Levy walks in quenched one-dimensional random and fractal quasi-lattices, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling properties of the random-walk probability distribution, we show that the effect of the varying scattering length can be reabsorbed in the multiplicative coefficient of the scaling length. This leads to a superscaling behavior, where the dynamical exponents and also the scaling functions do not depend on the value of the scattering length. Within the scaling framework, we obtain an exact expression for the multiplicative coefficient as a function of the scattering length both in the annealed and in the quenched random and fractal cases. Our analytic results are compared with numerical simulations, with excellent agreement, and are supposed to hold also in higher dimensions

Quantum Criticality in a Bosonic Josephson Junction

In this paper we consider a bosonic Josephson junction described by a two-mode Bose-Hubbard model, and we thoroughly analyze a quantum phase transition occurring in the system in the limit of infinite bosonic population. We discuss the relation between this quantum phase transition and the dynamical bifurcation occurring in the spectrum of the Discrete Self Trapping equations describing the system at the semiclassical level. In particular, we identify five regimes depending on the strength of the effective interaction among bosons, and study the finite-size effects arising from the finiteness of the bosonic population. We devote a special attention to the critical regime which reduces to the dynamical bifurcation point in the thermodynamic limit of infinite bosonic population. Specifically, we highlight an anomalous scaling in the population imbalance between the two wells of the trapping potential, as well as in two quantities borrowed from Quantum Information Theory, i.e. the entropy of entanglement and the ground-state fidelity. Our analysis is not limited to the zero temperature case, but considers thermal effects as well.

Non-equilibrium critical properties of the Ising model on product graphs

We study numerically the non-equilibrium critical properties of the Ising model defined on direct products of graphs, obtained from factor graphs without phase transition (Tc = 0). On this class of product graphs, the Ising model features a finite temperature phase transition, and we find a pattern of scaling behaviors analogous to the one known on regular lattices: Observables take a scaling form in terms of a function L(t) of time, with the meaning of a growing length inside which a coherent fractal structure, the critical state, is progressively formed. Computing universal quantities, such as the critical exponents and the limiting fluctuation-dissipation ratio X_infty, allows us to comment on the possibility to extend universality concepts to the critical behavior on inhomogeneous substrates.

Local and average behavior in inhomogeneous superdiffusive media

We consider a random walk on one-dimensional inhomogeneous graphs built from Cantor fractals. Our study is motivated by recent experiments that demonstrated superdiffusion of light in complex disordered materials, thereby termed Levy glasses. We introduce a geometric parameter $alpha$ which plays a role analogous to the exponent characterizing the step length distribution in random systems. We study the large-time behavior of both local and average observables; for the latter case, we distinguish two different types of averages, respectively over the set of all initial sites and over the scattering sites only. The single long jump approximation is applied to analytically determine the different asymptotic behaviours as a function of $alpha$ and to understand their origin. We also discuss the possibility that the root of the mean square displacement and the characteristic length of the walker distribution may grow according to different power laws; this anomalous behaviour is typical of processes characterized by Levy statistics and here, in particular, it is shown to influence average quantities.

Phase ordering and universality for continuous symmetry models on graphs

We study the phase-ordering kinetics following a temperature quench of O(N) continuous symmetry models with and 4 on graphs. By means of extensive simulations, we show that the global pattern of scaling behaviours is analogous to the one found on usual lattices. The exponent a for the integrated response function and the exponent z, describing the growing length, are related to the large scale topology of the networks through the spectral dimension and the fractal dimension alone, by means of the same expressions as are provided by the analytic solution of the inifnite N limit. This suggests that the large N value of these exponents could be exact for every N.