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arXiv2013

Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks

We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps η_i's of the RW, f(η), is symmetric and its Fourier transform has the small k behavior 1-\hat{f}(k)\sim| k|^μwith 0 < μ\leq 2. We compute the joint probability density function (pdf) P_n(g,l) of G_n and L_n and show that, when n \to \infty, it approaches a limiting pdf p(g,l). The corresponding marginal pdf of the gap, p_{\rm gap}(g), is found to behave like p_{\rm gap}(g) \sim g^{-1 - μ} for g \gg 1 and 0<μ< 2. We show that the limiting marginal distribution of L_n, p_{\rm time}(l), has an algebraic tail p_{\rm time}(l) \sim l^{-γ(μ)} for l \gg 1 with γ(1<μ\leq 2) = 1 + 1/μ, and γ(0<μ<1) = 2. For l, g \gg 1 with fixed l g^{-μ}, p(g,l) takes the scaling form p(g,l) \sim g^{-1-2μ} \tilde p_μ(l g^{-μ}) where \tilde p_μ(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.

cond-mat.stat-mech↗

arXiv2013

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

We investigate the conditions required for general spin systems with frustration and disorder to display self-organized criticality, a property which so far has been established only for the fully-connected infinite-range Sherrington-Kirkpatrick Ising spin-glass model [Phys. Rev. Lett. 83, 1034 (1999)]. Here we study both avalanche and magnetization jump distributions triggered by an external magnetic field, as well as internal field distributions in the short-range Edwards-Anderson Ising spin glass for various space dimensions between 2 and 8, as well as the fixed-connectivity mean-field Viana-Bray model. Our numerical results, obtained on systems of unprecedented size, demonstrate that self-organized criticality is recovered only in the strict limit of a diverging number of neighbors, and is not a generic property of spin-glass models in finite space dimensions.

cond-mat.dis-nn↗

arXiv2013

Eigenvalue Spectra of Modular Networks

A large variety of dynamical processes that take place on networks can be expressed in terms of the spectral properties of some linear operator which reflects how the dynamical rules depend on the network topology. Often such spectral features are theoretically obtained by considering only local node properties, such as degree distributions. Many networks, however, possess large-scale modular structures that can drastically influence their spectral characteristics, and which are neglected in such simplified descriptions. Here we obtain in a unified fashion the spectrum of a large family of operators, including the adjacency, Laplacian and normalized Laplacian matrices, for networks with generic modular structure, in the limit of large degrees. We focus on the conditions necessary for the merging of the isolated eigenvalues with the continuous band of the spectrum, after which the planted modular structure can no longer be easily detected by spectral methods. This is a crucial transition point which determines when a modular structure is strong enough to affect a given dynamical process. We show that this transition happens in general at different points for the different matrices, and hence the detectability threshold can vary significantly depending on the operator chosen. Equivalently, the sensitivity to the modular structure of the different dynamical processes associated with each matrix will be different, given the same large-scale structure present in the network. Furthermore, we show that, with the exception of the Laplacian matrix, the different transitions coalesce into the same point for the special case where the modules are homogeneous, but separate otherwise.

physics.data-an↗

arXiv2013

Transport of a Bose gas in 1D disordered lattices at the fluid-insulator transition

We investigate the momentum-dependent transport of 1D quasi-condensates in quasiperiodic optical lattices. We observe a sharp crossover from a weakly dissipative regime to a strongly unstable one at a disorder-dependent critical momentum. In the limit of non-disordered lattices the observations indicate a contribution of quantum phase slips to the dissipation. We identify a set of critical disorder and interaction strengths for which such critical momentum vanishes, separating a fluid regime from an insulating one. We relate our observation to the predicted zero-temperature superfluid-Bose glass transition.

cond-mat.dis-nn↗

arXiv2013

Local conservation laws and the structure of the many-body localized states

We construct a complete set of local integrals of motion that characterize the many-body localized (MBL) phase. Our approach relies on the assumption that local perturbations act locally on the eigenstates in the MBL phase, which is supported by numerical simulations of the random-field XXZ spin chain. We describe the structure of the eigenstates in the MBL phase and discuss the implications of local conservation laws for its non-equilibrium quantum dynamics. We argue that the many-body localization can be used to protect coherence in the system by suppressing relaxation between eigenstates with different local integrals of motion.

cond-mat.dis-nn↗

arXiv2013

Emergent Spin Excitations in a Bethe Lattice at Percolation

We study the spin 1/2 quantum Heisenberg antiferromagnet on a Bethe lattice diluted to the percolation threshold. Dilution creates areas of even/odd sublattice imbalance resulting in "dangling spins" (L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006); Phys. Rev. B 81, 054417 (2010)). These collectively act as "emergent" spin 1/2 degrees of freedom and are responsible for the creation of a set of low lying "quasidegenerate states". Using Density Matrix Renormalization Group (DMRG) calculations, we detect the presence and location of these emergent spins. We find an effective Hamiltonian of these emergent spins, with Heisenberg interactions that decay exponentially with the distance between them.

cond-mat.str-el↗

arXiv2013

Bose glass transition and spin-wave localization for 2D bosons in a random potential

A spin-wave approach of the zero temperature superfluid-insulator transition for two-dimensional hard-core bosons in a random potential $μ=\pm$ W is developed. While at the classical level there is no intervening phase between the Bose-condensed superfluid (SF) and the gapped disordered insulator, the introduction of quantum fluctuations leads to a much richer physics. Upon increasing the disorder strength W, the Bose-condensed fraction disappears first, before the SF. Then a gapless Bose-glass phase emerges over a finite region until the insulator appears. Furthermore, in the strongly disordered SF regime, a mobility edge in the spin-wave excitation spectrum is found at a finite frequency $Ω_c$ decreasing with W, and presumably vanishing in the Bose-glass phase.

cond-mat.str-el↗

arXiv2013

Novel disordering mechanism in ferromagnetic systems with competing interactions

Ferromagnetic Ising systems with competing interactions are considered in the presence of a random field. We find that in three space dimensions the ferromagnetic phase is disordered by a random field which is considerably smaller than the typical interaction strength between the spins. This is the result of a novel disordering mechanism triggered by an underlying spin-glass phase. Calculations for the specific case of the long-range dipolar LiHo_xY_{1-x}F_4 compound suggest that the above mechanism is responsible for the peculiar dependence of the critical temperature on the strength of the random field and the broadening of the susceptibility peaks as temperature is decreased, as found in recent experiments by Silevitch et al. [Nature (London) 448, 567 (2007)]. Our results thus emphasize the need to go beyond the standard Imry-Ma argument when studying general random-field systems.

cond-mat.dis-nn↗

arXiv2013

Experimental Observation of a Fundamental Length Scale of Waves in Random Media

Waves propagating through a weakly scattering random medium show a pronounced branching of the flow accompanied by the formation of freak waves, i.e., extremely intense waves. Theory predicts that this strong fluctuation regime is accompanied by its own fundamental length scale of transport in random media, parametrically different from the mean free path or the localization length. We show numerically how the scintillation index can be used to assess the scaling behavior of the branching length. We report the experimental observation of this scaling using microwave transport experiments in quasi-two-dimensional resonators with randomly distributed weak scatterers. Remarkably, the scaling range extends much further than expected from random caustics statistics.

cond-mat.dis-nn↗

arXiv2013

Cooperativity and the freezing of molecular motion at the glass transition

The slowing down of molecular dynamics when approaching the glass transition generally proceeds much stronger than expected for thermally activated motions. This strange phenomenon can be formally ascribed to a temperature-dependent activation energy E(T). In the present work, via measurements of the third-order nonlinear dielectric susceptibility, we deduce the increase of the number of correlated molecules Ncorr when approaching the glass transition and find a surprisingly simple correlation of E(T) and Ncorr(T). This provides strong evidence that the non-canonical temperature development of glassy dynamics is caused by a temperature-dependent energy barrier arising from the cooperative motion of ever larger numbers of molecules at low temperatures.

cond-mat.soft↗

arXiv2013

Superfluid Stiffness of a Driven Dissipative Condensate with Disorder

Observations of macroscopic quantum coherence in driven systems, e.g. polariton condensates, have strongly stimulated experimental as well as theoretical efforts during the last decade. We address the question of whether a driven quantum condensate is a superfluid, allowing for the effects of disorder and its non-equilibrium nature. We predict that for spatial dimensions d<4 the superfluid stiffness vanishes once the condensate exceeds a critical size, and treat in detail the case d=2. Thus a non-equilibrium condensate is not a superfluid in the thermodynamic limit, even for weak disorder, although superfluid behavior would persist in small systems.

cond-mat.quant-gas↗

arXiv2013

Near-extreme statistics of Brownian motion

We study the statistics of near-extreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states (DOS) near the maximum ρ(r,t) which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function (PDF) of ρ(r,t) and obtain an explicit expression for the moments, \langle [ρ(r,t)]^k \rangle, for arbitrary integer k. We also study near-extremes of constrained BM, like the Brownian bridge. Finally we also present numerical simulations to check our analytical results.

cond-mat.stat-mech↗

arXiv2013

Density of States Scaling at the Semimetal to Metal Transition in Three Dimensional Topological Insulators

The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of disordered $\mathbb{Z}_2$ topological insulator as an important example, we compute the single particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent $ν$ and the dynamical critical exponent $z$ are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed.

cond-mat.mes-hall↗

arXiv2013

Position-dependent diffusion of light in disordered waveguides

Diffusion has been widely used to describe a random walk of particles or waves, and it requires only one parameter -- the diffusion constant. For waves, however, diffusion is an approximation that disregards the possibility of interference. Anderson localization, which manifests itself through a vanishing diffusion coefficient in an infinite system, originates from constructive interference of waves traveling in loop trajectories -- pairs of time-reversed paths returning to the same point. In an open system of finite size, the return probability through such paths is reduced, particularly near the boundary where waves may escape. Based on this argument, the self-consistent theory of localization and the supersymmetric field theory predict that the diffusion coefficient varies spatially inside the system. A direct experimental observation of this effect is a challenge because it requires monitoring wave transport inside the system. Here, we fabricate two-dimensional photonic random media and probe position-dependent diffusion inside the sample from the third dimension. By varying the geometry of the system or the dissipation which also limits the size of loop trajectories, we are able to control the renormalization of the diffusion coefficient. This work shows the possibility of manipulating diffusion via the interplay of localization and dissipation.

physics.optics↗

arXiv2014

Absence of Anderson localization of light in a random ensemble of point scatterers

As discovered by Philip Anderson in 1958, strong disorder can block propagation of waves and lead to the localization of wave-like excitations in space. Anderson localization of light is particularly exciting in view of its possible applications for random lasing or quantum information processing. We show that, surprisingly, Anderson localization of light cannot be achieved in a random three-dimensional ensemble of point scattering centers that is the simplest and widespread model to study the multiple scattering of waves. Localization is recovered if the vector character of light is neglected. This shows that, at least for point scatterers, the polarization of light plays an important role in the Anderson localization problem.

physics.optics↗

arXiv2013

Persistent memory in athermal systems in deformable energy landscapes

We show that memory can be encoded in a model amorphous solid subjected to athermal oscillatory shear deformations, and in an analogous spin model with disordered interactions, sharing the feature of a deformable energy landscape. When these systems are subjected to oscillatory shear deformation, they retain memory of the deformation amplitude imposed in the training phase, when the amplitude is below a "localization" threshold. Remarkably, multiple, persistent, memories can be stored using such an athermal, noise-free, protocol. The possibility of such memory is shown to be linked to the presence of plastic deformations and associated limit cycles traversed by the system, which exhibit avalanche statistics also seen in related contexts.

cond-mat.stat-mech↗

arXiv2013

Quantum Hall criticality and localization in graphene with short-range impurities at the Dirac point

We explore the longitudinal conductivity of graphene at the Dirac point in a strong magnetic field with two types of short-range scatterers: adatoms that mix the valleys and "scalar" impurities that do not mix them. A scattering theory for the Dirac equation is employed to express the conductance of a graphene sample as a function of impurity coordinates; an averaging over impurity positions is then performed numerically. The conductivity $σ$ is equal to the ballistic value $4e^2/πh$ for each disorder realization provided the number of flux quanta considerably exceeds the number of impurities. For weaker fields, the conductivity in the presence of scalar impurities scales to the quantum-Hall critical point with $σ\simeq 4 \times 0.4 e^2/h$ at half filling or to zero away from half filling due to the onset of Anderson localization. For adatoms, the localization behavior is obtained also at half filling due to splitting of the critical energy by intervalley scattering. Our results reveal a complex scaling flow governed by fixed points of different symmetry classes: remarkably, all key manifestations of Anderson localization and criticality in two dimensions are observed numerically in a single setup.

cond-mat.mes-hall↗

arXiv2013

Disorder-induced regular dynamics in oscillating lattices

We explore the impact of weak disorder on the dynamics of classical particles in a periodically oscillating lattice. It is demonstrated that the disorder induces a hopping process from diffusive to regular motion i.e. we observe the counterintuitive phenomenon that disorder leads to regular behaviour. If the disorder is localized in a finite-sized part of the lattice, the described hopping causes initially diffusive particles to even accumulate in regular structures of the corresponding phase space. A hallmark of this accumulation is the emergence of pronounced peaks in the velocity distribution of particles which should be detectable in state of the art experiments e.g. with cold atoms in optical lattices.

nlin.CD↗