Ordered and disordered dynamics in monolayers of rolling particles

We consider the ordered and disordered dynamics for monolayers of rolling self-interacting particles modeling water molecules. The rolling constraint represents a simplified model of a strong, but rapidly decaying bond with the surface. We show the existence and nonlinear stability of ordered lattice states, as well as disturbance propagation through and chaotic vibrations of these states. We study the dynamics of disordered gas states and show that there is a surprising and universal linear connection between distributions of angular and linear velocity, allowing definition of temperature.     

Effect of evanescent channels on position-dependent diffusion in disordered waveguides

Abstract

We employ ab initio simulations of wave transport in disordered waveguides to demonstrate explicitly that although accounting for evanescent channels manifests itself in the renormalization of the transport mean free path, the position-dependent diffusion coefficient, as well as distributions of angular transmission, total transmission and conductance, all remain universal.     

Minimum spanning trees on random networks

We show that the geometry of minimum spanning trees (MST) on random graphs is universal. Because of this geometric universality, we are able to characterize the energy of MST using a scaling distribution [P(epsilon)] found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to P(epsilon). We discuss the relationship to invasion percolation, to the directed polymer in a random media, to uniform spanning trees, and also the implications for the broader issue of universality in disordered systems.     

Universal behavior of optimal paths in weighted networks with general disorder

We study the statistics of the optimal path in both random and scale-free networks, where weights are taken from a general distribution P(w). We find that different types of disorder lead to the same universal behavior. Specifically, we find that a single parameter (S defined as AL(-1/v) for d-dimensional lattices, and S defined as AN(-1/3) for random networks) determines the distributions of the optimal path length, including both strong and weak disorder regimes. Here v is the percolation connectivity exponent, and A depends on the percolation threshold and P(w). We show that for a uniform P(w), Poisson or Gaussian, the crossover from weak to strong does not occur, and only weak disorder exists.     

Statistical description of eigenfunctions in chaotic and weakly disordered systems beyond universality

We present a semiclassical approach to eigenfunction statistics in chaotic and weakly disordered quantum systems which goes beyond random matrix theory, supersymmetry techniques, and existing semiclassical methods. The approach is based on a generalization of Berry's random wave model, combined with a consistent semiclassical representation of spatial two-point correlations. We derive closed expressions for arbitrary wave-function averages in terms of universal coefficients and sums over classical paths, which contain, besides the supersymmetry results, novel oscillatory contributions. Their physical relevance is demonstrated in the context of Coulomb blockade physics.     

Correlations due to localization in quantum eigenfunctions of disordered microwave cavities

Statistical properties of experimental eigenfunctions of quantum chaotic and disordered microwave cavities are shown to demonstrate nonuniversal correlations due to localization. Varying energy E and mean free path l enable us to experimentally tune from localized to delocalized states. Large level-to-level inverse participation ratio ( I2) fluctuations are observed for the disordered billiards, whose distribution is strongly asymmetric about . The spatial density autocorrelations of eigenfunctions are shown to spatially decay exponentially and the decay lengths are experimentally determined. All the results are quantitatively consistent with calculations based upon nonlinear sigma models.     

Crystal-field induced dipoles in heteropolar crystals II: Physical significance

The significance of dipole moments induced by crystal fields in heteropolar crystals is discussed with respect to some aspects of solid state physics. Experimental results from structural analyses that provide data on induced dipoles are summarized. The concept of ionic radii is reconsidered, and a new tabulation scheme is proposed in terms of deformed charge distributions. It is shown that spontaneous polarization as well as the pyro- and piezoelectric coefficients are not independent sets of crystallographic constants, but are accounted for by the structural parameters, the ionic polarizabilities and the elastic constants. The dipole concept is extended to statistically induced or random dipoles. They can account for an important part of the binding energy of substitutionally disordered and non-stoichiometric compounds and, therefore, are concluded to stabilize disorder in solids.     

Spectral correlations in systems undergoing a transition from periodicity to disorder

We study the spectral statistics for extended yet finite quasi-one-dimensional systems, which undergo a transition from periodicity to disorder. In particular, we compute the spectral two-point form factor, and the resulting expression depends on the degree of disorder. It interpolates smoothly between the two extreme limits-the approach to Poissonian statistics in the (weakly) disordered case, and the universal expressions derived in T. Dittrich, B. Mehlig, H. Schanz, and U. Smilansky, Chaos Solitons Fractals 8, 1205 (1997); Phys. Rev. E 57, 359 (1998); B. D. Simons and B. L. Altshuler, Phys. Rev. Lett. 70, 4063 (1993); and N. Taniguchi and B. L. Altshuler, ibid. 71, 4031 (1993) for the periodic case. The theoretical results agree very well with the spectral statistics obtained numerically for chains of chaotic billiards and graphs.     

Universal features of terahertz absorption in disordered materials

Using an analytical theory, experimental terahertz time-domain spectroscopy data, and numerical evidence, we demonstrate that the frequency dependence of the absorption coupling coefficient between far-infrared photons and atomic vibrations in disordered materials has the universal functional form, C(omega)=A+Bomega(2), where the material-specific constants A and B are related to the distributions of fluctuating charges obeying global and local charge neutrality, respectively.     

Optimal paths in disordered media: scaling of the crossover from self-similar to self-affine behavior

We study optimal paths in disordered energy landscapes using energy distributions of the type P(log(10) E)=const that lead to the strong disorder limit. If we truncate the distribution, so that P(log(10) E)=const only for E(min) < or =E < or =E(max), and P(log(10) E)=0 otherwise, we obtain a crossover from self-similar (strong disorder) to self-affine (moderate disorder) behavior at a path length l(x). We find that l(x) proportional, variant[log(10)(E(max)/E(min))](kappa), where the exponent kappa has the value kappa=1.60 +/- 0.03 both in d=2 and d=3. We show how the crossover can be understood from the distribution of local energies on the optimal paths.     

Disordered solids: Universal behavior of structure, dynamics, and transport phenomena

Disordered materials (glasses and amorphous substances, melts, polymers, biological media, etc.) are an important class of objects. Despite the chaos usually associated with their structure, glasses and amorphous substances of various kinds (semiconducting, dielectric, metallic) possess a universal spatial scale length ∼1 nm, an order parameter, which can be as important theoretically as the unit cell for crystals. The disorder in disordered substances is not absolute; the periodicity positions of atomic inherent in crystals is maintained within a few coordination spheres and is then somehow destroyed. The way in which the order breaks down makes it possible to distinguish the glasses from amorphous materials in terms of the form of the structural correlation function. The inhomogeneities in question are not exotic, unique formations or analogs of defects in crystals, but are the fragments out of which amorphous substances and glasses are entirely constructed. The spatial inhomogeneity of disordered substances having a characteristic scale length of ∼1 nm leads to some universal characteristics in their vibrational properties, changes the relaxation mechanism for electronic excitation, and determines the specific features of charge transport. Fiz. Tverd. Tela (St. Petersburg) 41, 805–808 (May 1999)     

The effect of determined chaos on light collection in optical systems

In terms of the dynamic approach, the collection of light in optical systems and the influence of determined chaos on the photometry and fluctuations of regular and chaotic collection are considered. The photometric relationships generalizing the formula of the integrating sphere as applied to chaotic collection are obtained. A universal law for noise in the regular light collection is predicted and found to be in good agreement with the available experimental data. The relationships studied can find use in the elaboration of a new-design detectors, light guides, light-emitting diodes, etc., for the enhancement of their efficiency and the reduction of noise.     

Stochastic path integral formulation of full counting statistics

We derive a stochastic path integral representation of counting statistics in semiclassical systems. The formalism is introduced on the simple case of a single chaotic cavity with two quantum point contacts, and then further generalized to find the propagator for charge distributions with an arbitrary number of counting fields and generalized charges. The counting statistics is given by the saddle-point approximation to the path integral, and fluctuations around the saddle point are suppressed in the semiclassical approximation. We use this approach to derive the current cumulants of a chaotic cavity in the hot-electron regime.     

A path integral approach to effective non-linear medium

In this article, we propose a new method to compute the effective properties of non-linear disordered media. We use the fact that the effective constants can be defined through the minimum of an energy functional. We express this minimum in terms of a path integral allowing us to use many-body techniques. We obtain the perturbation expansion of the effective constants to second order in disorder, for any kind of non-linearity. We apply our method to the case of strong non-linearities (i.e. , where is fluctuating from point to point), and to the case of weak non-linearity (i.e. where and fluctuate from point to point). Our results are in agreement with previous ones, and could be easily extended to other types of non-linear problems in disordered systems. Received: 13 May 1998 / Accepted: 27 July 1998     

Testing statistical bounds on entanglement using quantum chaos

Previous results indicate that while chaos can lead to substantial entropy production, thereby maximizing dynamical entanglement, this still falls short of maximality. Random matrix theory modeling of composite quantum systems, investigated recently, entails a universal distribution of the eigenvalues of the reduced density matrices. We demonstrate that these distributions are realized in quantized chaotic systems by using a model of two coupled and kicked tops. We derive an explicit statistical universal bound on entanglement, which is also valid for the case of unequal dimensionality of the Hilbert spaces involved, and show that this describes well the bounds observed using composite quantized chaotic systems such as coupled tops.     

Decay of a thermofield-double state in chaotic quantum systems

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite N for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.     

Exact asymptotic behavior of correlation functions for disordered spin-1/2 XXZ chains

We consider an XXZ spin-1/2 chain in the presence of several types of disorder that do not break the XY symmetry of the system. We calculate the complete asymptotic form of the spin-correlation functions at zero temperature at the transition between liquid and disordered phase that occurs for a special value of anisotropy in the limit of small disorder. Apart from a universal power law decay of correlations, we find additional logarithmic corrections due to marginally irrelevant operator of disorder.     

Energy levels and their correlations in quasicrystals

Quasicrystals can be considered, from the point of view of their electronic properties, as being intermediate between metals and insulators. For example, experiments show that quasicrystalline alloys such as AlCuFe or AlPdMn have conductivities far smaller than those of the metals that these alloys are composed from. Wavefunctions in a quasicrystal are typically intermediate in character between the extended states of a crystal and the exponentially localized states in the insulating phase, and this is also reflected in the energy spectrum and the density of states. In the theoretical studies we consider in this review, the quasicrystals are described by a pure hopping tight binding model on simple tilings. We focus on spectral properties, which we compare with those of other complex systems, in particular, the Anderson model of a disordered metal. We discuss ‘strong‘ and ‘weak’ quasicrystals, which are described by different universal laws. We find similarities and universal behaviour, but also significant differences between quasiperiodic models and models with disorder. Like weakly disordered metals, the quasicrystal can be described by the universal level statistics that can be derived from random matrix theory. These level statistics are only one aspect of the energy spectrum, whose very large fluctuations can also be described by a level spacing distribution that is log-normal. An analysis of spectral rigidity shows that electrons diffuse with a bigger exponent (super-diffusion) than in a disordered metal. Adding disorder attenuates the singular properties of the perfect quasicrystal, and leads to improved transport. Spectral properties are also used in computing conductances of such systems, and to attempt to resolve the experimental enigmas such as whether quasicrystals are intrinsically conductors, and if so, how conductances depend on the structure.     

Bosonization for disordered and chaotic systems

Using a supersymmetry formalism, we reduce exactly the problem of electron motion in an external potential to a new supermatrix model valid at all distances. All approximate nonlinear sigma models obtained previously for disordered systems can be derived from our exact model using a coarse-graining procedure. As an example, we consider a model for a smooth disorder and demonstrate that using our approach does not lead to a "mode-locking" problem. As a new application, we consider scattering on strong impurities for which the Born approximation cannot be used. Our method provides a new calculational scheme for disordered and chaotic systems.