arXiv subjects

J. A. Mendez-Bermudez

Publications and source records attributed to J. A. Mendez-Bermudez.

At least 37 records · Page 2Linked to original sources

arXiv2019

Spectral and localization properties of random bipartite graphs

Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of $n$ nodes that is decomposed into two disjoint subsets, having $m$ and $n-m$ vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter $α\in[0,1]$ that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter $ξ\equivξ(n,m,α)$ that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when $ξ<1/10$ ($ξ>10$) the eigenvectors are localized (extended), whereas the localization--to--delocalization transition occurs in the interval $1/10<ξ<10$. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed $ξ$, the spectral properties of our graph model are also universal.

cond-mat.dis-nn↗

arXiv2019

Non-standard anomalous heat conduction in harmonic chains with correlated isotopic disorder

We address the general problem of heat conduction in one dimensional harmonic chain, with correlated isotopic disorder, attached at its ends to white noise or oscillator heat baths. When the low wavelength $μ$ behavior of the power spectrum $W$ (of the fluctuations of the random masses around their common mean value) scales as $W(μ)\sim μ^β$, the asymptotic thermal conductivity $κ$ scales with the system size $N$ as $κ\sim N^{(1+β)/(2+β)}$ for free boundary conditions, whereas for fixed boundary conditions $κ\sim N^{(β-1)/(2+β)}$; where $β>-1$, which is the usual power law scaling for one dimensional systems. Nevertheless, if $W$ does not scale as a power law in the low wavelength limit, the thermal conductivity may not scale in its usual form $κ\sim N^α$, where the value of $α$ depends on the particular one dimensional model. As an example of the latter statement, if $W(μ)\sim \exp(-1/μ)/μ^2$, $κ\sim N/(\log N)^3$ for fixed boundary conditions and $κ\sim N/\log(N)$ for free boundary conditions, which represent non-standard scalings of the thermal conductivity.

cond-mat.dis-nn↗

arXiv2019

Heat conduction in harmonic chains with Levy-type disorder

We consider heat transport in one-dimensional harmonic chains attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation $d$ between any two successive impurities is randomly distributed according to a power-law distribution $P(d)\sim 1/d^{α+1}$, being $α>0$. In the regime where the first moment of the distribution is well defined ($1<α<2$) the thermal conductivity $κ$ scales with the system size $N$ as $κ\sim N^{(α-3)/α}$ for fixed boundary conditions, whereas for free boundary conditions $κ\sim N^{(α-1)/α}$ if $N\gg1$. When $α=2$, the inverse localization length $λ$ scales with the frequency $ω$ as $λ\sim ω^2 \ln ω$ in the low frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a non-closed form. When $α>2$, the thermal conductivity scales as in the uncorrelated disorder case. The situation $α<1$ is only analyzed numerically, where $λ(ω)\sim ω^{2-α}$ which leads to the following asymptotic thermal conductivity: $κ\sim N^{-(α+1)/(2-α)}$ for fixed boundary conditions and $κ\sim N^{(1-α)/(2-α)}$ for free boundary conditions.

cond-mat.dis-nn↗

arXiv2019

Real and imaginary energy gaps: a comparison between single excitation Superradiance and Superconductivity and robustness to disorder

A comparison between the single particle spectrum of the discrete Bardeen-Cooper-Schrieffer (BCS) model, used for small superconducting grains, and the spectrum of a paradigmatic model of Single Excitation Superradiance (SES) is presented. They are both characterized by an equally spaced energy spectrum (Picket Fence) where all the levels are coupled between each other by a constant coupling which is real for the BCS model and purely imaginary for the SES model. While the former corresponds to the discrete BCS-model describing the coupling of Cooper pairs in momentum space and it induces a Superconductive regime, the latter describes the coupling of single particle energy levels to a common decay channel and it induces a Superradiant transition. We show that the transition to a Superradiant regime can be connected to the emergence of an imaginary energy gap, similarly to the transition to a Superconductive regime where a real energy gap emerges. Despite their different physical origin, it is possible to show that both the Superradiant and the Superconducting gaps have the same magnitude in the large gap limit. Nevertheless, some differences appear: while the critical coupling at which the Superradiant gap appears is independent of the system size $N$, for the Superconductivity gap it scales as $(\ln N)^{-1}$. The presence of a gap in the imaginary energy axis between the Superradiant and the Subradiant states shares many similarities with the "standard" gap on the real energy axis: the superradiant state is protected against disorder from the imaginary gap as well as the superconducting ground state is protected by the real energy gap. Moreover we connect the origin of the gapped phase to the long-range nature of the coupling between the energy levels.

cond-mat.supr-con↗

arXiv2017

Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of weighted random--geometric and random--rectangular graphs. A random--geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random--rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths $a$ and $1/a$, the connection radius $r$, and the number of vertices $N$. We then study in detail the case $a=1$ which corresponds to weighted RGGs and explore weighted RRGs characterized by $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg1$. In general we look for the scaling properties of $P(s)$ and $\ell$ as a function of $a$, $r$ and $N$. We find that the ratio $r/N^γ$, with $γ(a)\approx -1/2$, fixes the properties of both RGGs and RRGs. Moreover, when $a\ge 10$ we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio $r/{\cal C}N^γ$, with ${\cal C}\approx a$.

physics.soc-ph↗

arXiv2017

Diluted banded random matrices: Scaling behavior of eigenfunction and spectral properties

We demonstrate that the normalised localization length $β$ of the eigenfunctions of diluted (sparse) banded random matrices follows the scaling law $β=x^*/(1+x^*)$. The scaling parameter of the model is defined as $x^*\propto(b_{eff}^2/N)^δ$, where $b_{eff}$ is the average number of non-zero elements per matrix row, $N$ is the matrix size, and $δ\sim 1$. Additionally, we show that $x^*$ also scales the spectral properties of the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues.

cond-mat.dis-nn↗

arXiv2016

A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems

We perform a detailed numerical study of the conductance $G$ through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies $ε$ of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large $ε$, $P(ε)\sim 1/ε^{1+α}$ with $α\in(0,2)$. Our model serves as a generalization of 1D Lloyd's model, which corresponds to $α=1$. First, we verify that the ensemble average $\left\langle -\ln G\right\rangle$ is proportional to the length of the wire $L$ for all values of $α$, providing the localization length $ξ$ from $\left\langle-\ln G\right\rangle=2L/ξ$. Then, we show that the probability distribution function $P(G)$ is fully determined by the exponent $α$ and $\left\langle-\ln G\right\rangle$. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at $G=0$ and $1$. In addition, we show that $P(\ln G)$ is proportional to $G^β$, for $G\to 0$, with $β\leα/2$, in agreement to previous studies.

cond-mat.dis-nn↗

arXiv2015

Universality in the spectral and eigenfunction properties of random networks

By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) {\it fully} random networks are universal for fixed average degree $ξ\equiv αN$ ($α$ and $N$ being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well $P(s)$ in the transition from $α=0$, when the vertices in the network are isolated, to $α=1$, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with {\it diagonal disorder} also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

cond-mat.dis-nn↗

arXiv2014

The Lévy Map: A two-dimensional nonlinear map characterized by tunable Lévy flights

Once recognizing that point particles moving inside the extended version of the rippled billiard perform Lévy flights characterized by a Lévy-type distribution $P(\ell)\sim \ell^{-(1+α)}$ with $α=1$, we derive a generalized two-dimensional non-linear map $M_α$ able to produce Lévy flights described by $P(\ell)$ with $0<α<2$. Due to this property, we name $M_α$ as the Lévy Map. Then, by applying Chirikov's overlapping resonance criteria we are able to identify the onset of global chaos as a function of the parameters of the map. With this, we state the conditions under which the Lévy Map could be used as a Lévy pseudo-random number generator and, furthermore, confirm its applicability by computing scattering properties of disordered wires.

nlin.CD↗

arXiv2014

On the generalized dimensions of multifractal eigenstates

Recently, based on heuristic arguments, it was conjectured that an intimate relation exists between any multifractal dimensions, $D_q$ and $D_{q'}$, of the eigenstates of critical random matrix ensembles: $D_{q'} \approx qD_q[q'+(q-q')D_q]^{-1}$, $1\le q, q' \le 2$. Here, we verify this relation by extensive numerical calculations on critical random matrix ensembles and extend its applicability to $q<1/2$ but also to deterministic models producing multifractal eigenstates and to generic multifractal structures. We also demonstrate, for the scattering version of the power-law banded random matrix model at criticality, that the scaling exponents $σ_q$ of the inverse moments of Wigner delay times, $\bra τ_{\tbox W}^{-q} \ket \propto N^{-σ_q}$ where $N$ is the linear size of the system, are related to the level compressibility $χ$ as $σ_q\approx q(1-χ)[1+qχ]^{-1}$ for a limited range of $q$; thus providing a way to probe level correlations by means of scattering experiments.

cond-mat.dis-nn↗

arXiv2014

Transport through quasi-one-dimensional wires with correlated disorder

We study transport properties of bulk-disordered quasi-one-dimensional (Q1D) wires paying main attention to the role of long-range correlations embedded into the disorder. First, we show that for stratified disorder for which the disorder is the same for all individual chains forming the Q1D wire, the transport properties can be analytically described provided the disorder is weak. When the disorder in every chain is not the same, however, has the same binary correlator, the general theory is absent. Thus, we consider the case when only one channel is open and all others are closed. For this situation we suggest a semi-analytical approach which is quite effective for the description of the total transmission coefficient. Our numerical data confirm the validity of our approach. Such Q1D disordered structures with anomalous transport properties can be the subject of an experimental study.

cond-mat.dis-nn↗

arXiv2013

Dynamical properties of a dissipative discontinuous map: A scaling investigation

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action $\left< I^2 \right>$ as a function of the $n$-th iteration of the map as well as the parameters $K$ and $γ$, controlling nonlinearity and dissipation, respectively. We concentrate our efforts to study the case where the nonlinearity is large; i.e., $K\gg 1$. In this regime and for large initial action $I_0\gg K$, we prove that dissipation produces an exponential decay for the average action $\left< I \right>$. Also, for $I_0\cong 0$, we describe the behavior of $\left< I^2 \right>$ using a scaling function and analytically obtain critical exponents which are used to overlap different curves of $\left< I^2 \right>$ onto an universal plot. We complete our study with the analysis of the scaling properties of the deviation around the average action $ω$.

nlin.CD↗

arXiv2013

Scattering and transport properties of tight-binding random networks

We study numerically scattering and transport statistical properties of tight-binding random networks characterized by the number of nodes $N$ and the average connectivity $α$. We use a scattering approach to electronic transport and concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements $\bra |S_{mn}|^2 \ket$, the conductance probability distribution $w(T)$, the average conductance $\bra T \ket$, the shot noise power $P$, and the elastic enhancement factor $F$ by varying $α$ from small ($α\to 0$) to large ($α\to 1$) values. We also show that all these quantities are invariant for fixed $ξ=αN$. Moreover, we proposes a heuristic and universal relation between $\bra |S_{mn}|^2 \ket$, $\bra T \ket$, and $P$ and the disorder parameter $ξ$.

cond-mat.dis-nn↗

arXiv2013

Disorder to chaos transition in the conductance distribution of corrugated waveguides

We perform a detailed numerical study of the distribution of conductances $P(T)$ for quasi-one-dimensional corrugated waveguides as a function of the corrugation complexity (from rough to smooth). We verify the universality of $P(T)$ in both, the diffusive ($\bra T \ket> 1$) and the localized ($\bra T \ket\ll 1$) transport regimes. However, at the crossover regime ($\bra T \ket \sim 1$), we observe that $P(T)$ evolves from the surface-disorder to the bulk-disorder theoretical predictions for decreasing complexity in the waveguide boundaries. We explain this behavior as a transition from disorder to deterministic chaos; since, in the limit of smooth boundaries the corrugated waveguides are, effectively, linear chains of chaotic cavities.

cond-mat.dis-nn↗

arXiv2013

Photonic structures with disorder immunity

Periodic and disordered media are known to possess different transport properties, either classically or quantum-mechanically. This has been exhibited by effects such as Anderson localization in systems with disorder and the existence of photonic bandgaps in the periodic case. In this paper we analyze the transport properties of disordered waveguides with corners at very low frequencies, finding that the spectrum, conductance and wavefunctions are immune to disorder. Our waveguides are constructed by means of randomly oriented straight segments and connected by corners at right angles. Taking advantage of a trapping effect that manifests in the corner of a bent waveguide, we can show that a tight-binding approximation describes the system reasonably well for any degree of disorder. This provides a wide set of non-periodic geometries that preserve all the interesting transport properties of periodic media.

cond-mat.dis-nn↗

arXiv2012

Scaling properties of discontinuous maps

We study the scaling properties of discontinuous maps by analyzing the average value of the squared action variable $I^2$. We focus our study on two dynamical regimes separated by the critical value $K_c$ of the control parameter $K$: the slow diffusion ($K K_c$) regimes. We found that the scaling of $I^2$ for discontinuous maps when $K\ll K_c$ and $K\gg K_c$ obeys the same scaling laws, in the appropriate limits, than Chirikov's standard map in the regimes of weak and strong nonlinearity, respectively. However, due to absence of KAM tori, we observed in both regimes that $I^2\propto nK^β$ for $n\gg 1$ (being $n$ the $n$-th iteration of the map) with $β\approx 5/2$ when $K\ll K_c$ and $β\approx 2$ for $K\gg K_c$.

nlin.CD↗

arXiv2012

Photonic heterostructures with Levy-type disorder: statistics of coherent transmission

We study the electromagnetic transmission $T$ through one-dimensional (1D) photonic heterostructures whose random layer thicknesses follow a long-tailed distribution --Lévy-type distribution. Based on recent predictions made for 1D coherent transport with Lévy-type disorder, we show numerically that for a system of length $L$ (i) the average $<-\ln T> \propto L^α$ for $0<α<1$, while $<-\ln T> \propto L$ for $1\leα<2$, $α$ being the exponent of the power-law decay of the layer-thickness probability distribution; and (ii) the transmission distribution $P(T)$ is independent of the angle of incidence and frequency of the electromagnetic wave, but it is fully determined by the values of $α$ and $<\ln T>$.

physics.optics↗

arXiv2012

Multifractal dimensions for critical random matrix ensembles

Based on heuristic arguments we conjecture that an intimate relation exists between the eigenfunction multifractal dimensions $D_q$ of the eigenstates of critical random matrix ensembles $D_{q'} \approx qD_q[q'+(q-q')D_q]^{-1}$, $1\le q \le 2$. We verify this relation by extensive numerical calculations. We also demonstrate that the level compressibility $χ$ describing level correlations can be related to $D_q$ in a unified way as $D_q=(1-χ)[1+(q-1)χ]^{-1}$, thus generalizing existing relations with relevance to the disorder driven Anderson--transition.

cond-mat.dis-nn↗