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.

/pdf/statistical-mechanics-of-complex-networks-1zfjpvxipu.pdf

Statistical mechanics of complex networks

Metal-insulator transitions are accompanied by huge resistivity changes, even over tens of orders of magnitude, and are widely observed in condensed-matter systems. This article presents the observations and current understanding of the metal-insulator transition with a pedagogical introduction to the subject. Especially important are the transitions driven by correlation effects associated with the electron-electron interaction. The insulating phase caused by the correlation effects is categorized as the Mott Insulator. Near the transition point the metallic state shows fluctuations and orderings in the spin, charge, and orbital degrees of freedom. The properties of these metals are frequently quite different from those of ordinary metals, as measured by transport, optical, and magnetic probes. The review first describes theoretical approaches to the unusual metallic states and to the metal-insulator transition. The Fermi-liquid theory treats the correlations that can be adiabatically connected with the noninteracting picture. Strong-coupling models that do not require Fermi-liquid behavior have also been developed. Much work has also been done on the scaling theory of the transition. A central issue for this review is the evaluation of these approaches in simple theoretical systems such as the Hubbard model and $t\ensuremath{-}J$ models. Another key issue is strong competition among various orderings as in the interplay of spin and orbital fluctuations. Experimentally, the unusual properties of the metallic state near the insulating transition have been most extensively studied in $d$-electron systems. In particular, there is revived interest in transition-metal oxides, motivated by the epoch-making findings of high-temperature superconductivity in cuprates and colossal magnetoresistance in manganites. The article reviews the rich phenomena of anomalous metallicity, taking as examples Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Ru compounds. The diverse phenomena include strong spin and orbital fluctuations, mass renormalization effects, incoherence of charge dynamics, and phase transitions under control of key parameters such as band filling, bandwidth, and dimensionality. These parameters are experimentally varied by doping, pressure, chemical composition, and magnetic fields. Much of the observed behavior can be described by the current theory. Open questions and future problems are also extracted from comparison between experimental results and theoretical achievements.

Metal-insulator transitions

We provide a broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature dependent carrier transport in doped or gated graphene structures. A salient feature of our review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g. heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gap- less, massless, chiral Dirac spectrum are highlighted. Experiment and theory as well as quantum and semi-classical transport are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long- range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.

/pdf/electronic-transport-in-two-dimensional-graphene-32oou47sy4.pdf

Electronic transport in two-dimensional graphene

A revolution in electronics is in view, with the contemporary evolution of the two novel disciplines of spintronics and molecular electronics. A fundamental link between these two fields can be established using molecular magnetic materials and, in particular, single-molecule magnets. Here, we review the first progress in the resulting field, molecular spintronics, which will enable the manipulation of spin and charges in electronic devices containing one or more molecules. We discuss the advantages over more conventional materials, and the potential applications in information storage and processing. We also outline current challenges in the field, and propose convenient schemes to overcome them.

Molecular spintronics using single-molecule magnets

Shor and Grover demonstrated that a quantum computer can outperform any classical computer in factoring numbers1 and in searching a database2 by exploiting the parallelism of quantum mechanics. Whereas Shor's algorithm requires both superposition and entanglement of a many-particle system3, the superposition of single-particle quantum states is sufficient for Grover's algorithm4. Recently, the latter has been successfully implemented5 using Rydberg atoms. Here we propose an implementation of Grover's algorithm that uses molecular magnets6,7,8,9,10, which are solid-state systems with a large spin; their spin eigenstates make them natural candidates for single-particle systems. We show theoretically that molecular magnets can be used to build dense and efficient memory devices based on the Grover algorithm. In particular, one single crystal can serve as a storage unit of a dynamic random access memory device. Fast electron spin resonance pulses can be used to decode and read out stored numbers of up to 105, with access times as short as 10-10 seconds. We show that our proposal should be feasible using the molecular magnets Fe8 and Mn12.

Quantum computing in molecular magnets

Hall coefficient of insulating n-type CdSe.

In high-mobility silicon metal-oxide-semiconductor field-effect transistors, the ${g}^{*}{m}^{*}$ inferred indirectly from magnetoconductance and magnetoresistance measurements with the assumption that ${g}^{*}{\ensuremath{\mu}}_{B}{H}_{s}{=2E}_{F}$ are in surprisingly good agreement with ${g}^{*}{m}^{*}$ obtained by direct measurement of Shubnikov--de Haas oscillations. The enhanced susceptibility ${\ensuremath{\chi}}^{*}\ensuremath{\propto}{(g}^{*}{m}^{*})$ exhibits critical behavior of the form ${\ensuremath{\chi}}^{*}\ensuremath{\propto}(n\ensuremath{-}{n}_{0}{)}^{\ensuremath{-}\ensuremath{\alpha}}.$ We examine the significance of the field scale ${H}_{s}$ derived from transport measurements, and show that this field signals the onset of full spin polarization only in the absence of disorder. Our results suggest that disorder becomes increasingly important as the electron density is reduced toward the transition.

/pdf/spin-polarization-of-strongly-interacting-two-dimensional-19wcv33yc4.pdf

Spin polarization of strongly interacting two-dimensional electrons: The role of disorder

We report measurements of the magnetization of insulating phosphorus-doped silicon for fields up to 50 kG and temperatures down to 1.25 K, for dopant concentrations N=6.7\ifmmode\times\else\texttimes\fi{}${10}^{17}$ to 2.8\ifmmode\times\else\texttimes\fi{}${10}^{18}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$. The data are found to be in quantitative agreement with a generalization, to finite fields, of a scaling calculation of the magnetic properties of the insulating phase, and support a model of hierarchically coupled spin pairs.

Scaling behavior of the magnetization of insulating Si:P

Conductivity data for Si:B samples with dopant concentrations 1.01 ${n}_{c}lnl{1.22n}_{c}$ at temperatures between 0.07 and 0.5 K in magnetic fields from 0 to 9.0 T collapse onto a single universal curve $\ensuremath{\Delta}\ensuremath{\sigma}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}KT}^{1/2}F(H/T)$, the form expected for the magnetoconductance due to electron-electron interactions. This suggests that the metal-insulator transition is predominantly driven by electron correlations, and that localization, spin-flip scattering, and spin-orbit scattering are unimportant despite strong spin-orbit effects in Si:B.

Universal scaling of the magnetoconductance of metallic Si:B.

: The objective of these investigations was to study the magnetic properties of a collection of small magnetic particles or magnetized domains with special attention to detecting and documenting the occurence of macroscopic quantum tunneling of magnetization. Magnetized small particles are the basic elements used for magnetic storage, and our studies are important from a technological points of view: quantum-mechanical tunneling of the magnetization of small particles sets the ultimate limit on the reliability of magnetic storage and an upper limit on storage density. Experimentally, we succeeded in observing thermally assisted, field-tuned resonant tunneling between quantum spin states in a macroscopic-sized sample of the molecular magnetic material, manganese acetate. The magnetic properties of horse-spleen ferritin were also investigated. Theoretical studies included investigations of tunneling in random magnets, the introduction of the concept of a 'critical state', theoretical studies of antiferromagnetic grains with uncompensated sublattices, and the crossover from antiferromagnetic to ferromagnetic tunneling.

/pdf/quantum-tunneling-of-magnetization-1juwil2a8a.pdf

Quantum Tunneling of Magnetization.

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