There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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Statistical mechanics of complex networks

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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The Structure and Function of Complex Networks

Fundamentals of Plasmonics.- Electromagnetics of Metals.- Surface Plasmon Polaritons at Metal / Insulator Interfaces.- Excitation of Surface Plasmon Polaritons at Planar Interfaces.- Imaging Surface Plasmon Polariton Propagation.- Localized Surface Plasmons.- Electromagnetic Surface Modes at Low Frequencies.- Applications.- Plasmon Waveguides.- Transmission of Radiation Through Apertures and Films.- Enhancement of Emissive Processes and Nonlinearities.- Spectroscopy and Sensing.- Metamaterials and Imaging with Surface Plasmon Polaritons.- Concluding Remarks.

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Plasmonics: Fundamentals and Applications

Graphene’s success has shown that it is possible to create stable, single and few-atom-thick layers of van der Waals materials, and also that these materials can exhibit fascinating and technologically useful properties. Here we review the state-of-the-art of 2D materials beyond graphene. Initially, we will outline the different chemical classes of 2D materials and discuss the various strategies to prepare single-layer, few-layer, and multilayer assembly materials in solution, on substrates, and on the wafer scale. Additionally, we present an experimental guide for identifying and characterizing single-layer-thick materials, as well as outlining emerging techniques that yield both local and global information. We describe the differences that occur in the electronic structure between the bulk and the single layer and discuss various methods of tuning their electronic properties by manipulating the surface. Finally, we highlight the properties and advantages of single-, few-, and many-layer 2D materials in...

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Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene

Publisher Summary This chapter discusses several composites or granular or porous materials that display inhomogeneity on a macroscopic scale. In such materials, there are small, yet much larger than atomic, regions exhibiting macroscopic homogeneity. Different regions may display quite different properties. The chapter discusses the dc and ac electrical properties of composite media including the basic theory and results for dc electrical properties. It also presents a discussion of the effective-medium approximation, electrostatic resonances, exact bounds, and analytical properties and describes a number of static physical properties of composites. These properties include electrical conductivity and dielectric behavior near a percolation threshold, magnetotransport, thermoelectricity, superconductivity, and duality in two-dimensional composites. Finally, the chapter discusses the nonlinear properties and flicker noise in composites.

Physical Properties of Macroscopically Inhomogeneous Media

An old effective-medium approximation for the conductivity tensor of a randomly inhomogeneous medium is generalized to treat, in principle, materials consisting of crystallites of arbitrary shape and conductivity tensors of arbitrary symmetry. The effective-medium approximation is roughly analogous to the coherent-potential approximation (CPA) of alloy theory. The analog of the average-$t$-matrix approximation (ATA) is also formulated in a general way. The method is fully tractable analytically for ellipsoidal crystallites. Several applications are discussed. The effective conductivity of a polycrystal consisting of randomly oriented uniaxial crystallites is calculated as a function of the anisotropy of the grains. For a model polycrystal in an intense magnetic field, the CPA and ATA are compared, the former giving more accurate results.

Generalized effective-medium approach to the conductivity of an inhomogeneous material

The effective dielectric function ${\ensuremath{\epsilon}}_{e}$ for a medium of anisotropic inclusions embedded in an isotropic host is calculated using the Maxwell Garnett approximation. For uniaxial inclusions, ${\ensuremath{\epsilon}}_{e}$ depends on how well the inclusions are aligned. We apply this approximation to study ${\ensuremath{\epsilon}}_{e}$ for a model of quasi-one-dimensional organic polymers. The polymer is assumed to be made up of small single crystals embedded in an isotropic host of randomly oriented polymer chains. The host dielectric function is calculated using the effective-medium approximation (EMA). The resulting frequency-dependent ${\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ closely resembles experiment. Specifically, $\mathrm{Re}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ is negative over a wide frequency range, while $\mathrm{Im}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ exhibits a broad ``surface plasmon'' band at low frequencies, which results from localized electronic excitations within the crystallites. If the host is above the conductivity percolation threshold, $\mathrm{Im}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ has a low-frequency Drude peak in addition to the surface plasmon band, and $\mathrm{Re}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ is negative over an even wider frequency range. We also calculate the cubic nonlinear susceptibility ${\ensuremath{\chi}}_{e}(\ensuremath{\omega})$ of the polymer, using a nonlinear EMA. At certain frequencies, ${\ensuremath{\chi}}_{e}(\ensuremath{\omega})$ is found to be strongly enhanced above that of the corresponding single crystals. Our results suggest that the electromagnetic properties of conducting polymers can be understood by viewing the material as randomly inhomogeneous on a small scale such that the quasistatic limit is applicable.

https://www.physics.ohio-state.edu/~stroud/levy.pdf

Maxwell Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers

We present a theory for the diamagnetic response of weakly linked superconducting clusters. In the model, superconducting grains, each small compared to a penetration depth, are weakly coupled into closed loops. These support screening supercurrents in response to an external magnetic field. In a magnetic field, a large cluster can support many supercurrent-carrying states of nearly equal energy, but energy barriers between these states tend to inhibit hops from one state to another at low temperatures. The picture is similar to that often proposed for spin glasses. An important consequence is predicted to be a large difference between the dc and ac susceptibilities at low temperatures. The former, an equilibrium property, will fall off much more rapidly with field than the latter, which is generally a property of the metastable states. In addition, the magnetization of a cluster varies discontinuously with field; for a sufficiently large cluster, the magnetization is everywhere discontinuous. To check these conjectures, two examples are studied. The first consists of single loops of random areas and orientation, which can be solved analytically at zero temperature. The second involves random two-dimensional clusters of many closed loops, and is studied via careful Monte Carlo simulation at various temperaturesmore » and fields. Both examples display the expected strong differences between ac and dc susceptibilities at low temperatures. Our predictions are found to be quite similar to the experimental results of Bastuscheck et al. (Phys. Rev. B 24, 6707 (1981)) for the fibrous superconductors NbSe/sub 3/ and TaSe/sub 3/.« less

David Stroud

Papers

Physical Properties of Macroscopically Inhomogeneous Media

Generalized effective-medium approach to the conductivity of an inhomogeneous material

Maxwell Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers

Diamagnetic susceptibility of superconducting clusters: Spin-glass behavior.

Nonlinear susceptibilities of granular matter