Interaction induced delocalization of two particles: large system size calculations and dependence on interaction strength

The localization length $L_2$ of two interacting particles in a one-dimensional disordered system is studied for very large system sizes by two efficient and accurate variants of the Green function method. The numerical results (at the band center) can be well described by the functional form $L_2=L_1[0.5+c(U) L_1]$ where $L_1$ is the one-particle localization length and the coefficient $c(U)\approx 0.074 |U|/(1+|U|)$ depends on the strength $U$ of the on-site Hubbard interaction. The Breit-Wigner width or equivalently the (inverse) life time of non-interacting pair states is analytically calculated for small disorder and taking into account the energy dependence of the one-particle localization length. This provides a consistent theoretical explanation of the numerically found $U$-dependence of $c(U)$.Comment: 8 pages, 5 figures, LaTeX, EPJ macro package, submitted to the European Physical Journal

Eigenfunction structure and scaling of two interacting particles in the one-dimensional Anderson model

The localization properties of eigenfunctions for two interacting particles in the one-dimensional Anderson model are studied for system sizes up to $N=5000$ sites corresponding to a Hilbert space of dimension $\approx 10^7$ using the Green function Arnoldi method. The eigenfunction structure is illustrated in position, momentum and energy representation, the latter corresponding to an expansion in non-interacting product eigenfunctions. Different types of localization lengths are computed for parameter ranges in system size, disorder and interaction strengths inaccessible until now. We confirm that one-parameter scaling theory can be successfully applied provided that the condition of $N$ being significantly larger than the one-particle localization length $L_1$ is verified. The enhancement effect of the two-particle localization length $L_2$ behaving as $L_2\sim L_1^2$ is clearly confirmed for a certain quite large interval of optimal interactions strengths. Further new results for the interaction dependence in a very large interval, an energy value outside the band center, and different interaction ranges are obtained.Comment: 26 pages, 19 png and pdf figures, high quality gif files for panels of figures 1-4 are available at http://www.quantware.ups-tlse.fr/QWLIB/tipdisorder1d, final published version with minor corrections/revisions, addition of Journal reference and DO

Localization and absence of Breit-Wigner form for Cauchy random band matrices

We analytically calculate the local density of states for Cauchy random band matrices with strongly fluctuating diagonal elements. The Breit-Wigner form for ordinary band matrices is replaced by a Levy distribution of index $\mu=1/2$ and the characteristic energy scale $\alpha$ is strongly enhanced as compared to the Breit-Wigner width. The unperturbed eigenstates decay according to the non-exponential law $\propto e^{-\sqrt{\alpha t}}$. We analytically determine the localization length by a new method to derive the supersymmetric non-linear $\sigma$ model for this type of band matrices.Comment: 4 pages, 1 figur

Poisson statistics of PageRank probabilities of Twitter and Wikipedia networks

We use the methods of quantum chaos and Random Matrix Theory for analysis of statistical fluctuations of PageRank probabilities in directed networks. In this approach the effective energy levels are given by a logarithm of PageRank probability at a given node. After the standard energy level unfolding procedure we establish that the nearest spacing distribution of PageRank probabilities is described by the Poisson law typical for integrable quantum systems. Our studies are done for the Twitter network and three networks of Wikipedia editions in English, French and German. We argue that due to absence of level repulsion the PageRank order of nearby nodes can be easily interchanged. The obtained Poisson law implies that the nearby PageRank probabilities fluctuate as random independent variables.Comment: 4 pages, 4 figures http://www.quantware.ups-tlse.fr

Dynamical decoherence of a qubit coupled to a quantum dot or the SYK black hole

We study the dynamical decoherence of a qubit weakly coupled to a two-body random interaction model (TBRIM) describing a quantum dot of interacting fermions or the Sachdev-Ye-Kitaev (SYK) black hole model. We determine the rates of qubit relaxation and dephasing for regimes of dynamical thermalization of the quantum dot or of quantum chaos in the SYK model. These rates are found to correspond to the Fermi golden rule and quantum Zeno regimes depending on the qubit-fermion coupling strength. An unusual regime is found where these rates are practically independent of TBRIM parameters. We push forward an analogy between TBRIM and quantum small-world networks with an explosive spreading over exponentially large number of states in a finite time being similar to six degrees of separation in small-world social networks. We find that the SYK model has approximately two-three degrees of separation.Comment: 17 pages, 15 pdf-figure

Poincar\'e recurrences and Ulam method for the Chirikov standard map

We study numerically the statistics of Poincar\'e recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We also develop a new survival Monte Carlo method which allows us to study recurrences on times changing by ten orders of magnitude. We show that the recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. The values of Poincar\'e exponents of recurrences are determined for the two maps studied. We also discuss the localization properties of eigenstates of the Ulam matrix and their relation with the Poincar\'e recurrences.Comment: 11 pages, 14 figures, high resolution figures and video mpeg files available at: http://www.quantware.ups-tlse.fr/QWLIB/ulammethod

Freed by interaction kinetic states in the Harper model

We study the problem of two interacting particles in a one-dimensional quasiperiodic lattice of the Harper model. We show that a short or long range interaction between particles leads to emergence of delocalized pairs in the non-interacting localized phase. The properties of these Freed by Interaction Kinetic States (FIKS) are analyzed numerically including the advanced Arnoldi method. We find that the number of sites populated by FIKS pairs grows algebraically with the system size with the maximal exponent $b=1$, up to a largest lattice size $N=10946$ reached in our numerical simulations, thus corresponding to a complete delocalization of pairs. For delocalized FIKS pairs the spectral properties of such quasiperiodic operators represent a deep mathematical problem. We argue that FIKS pairs can be detected in the framework of recent cold atom experiments [M.~Schreiber {\it et al.} Science {\bf 349}, 842 (2015)] by a simple setup modification. We also discuss possible implications of FIKS pairs for electron transport in the regime of charge-density wave and high $T_c$ superconductivity.Comment: 26 pages, 21 pdf and png figures, additional data and high quality figures are available at http://www.quantware.ups-tlse.fr/QWLIB/fikspairs/ , parts of sections 2 and 3 moved to appendices, manuscript accepted for EPJ

Spectral properties of Google matrix of Wikipedia and other networks

We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.Comment: 10 pages, 9 figure