Multiferroics, defined for those multifunctional materials in which two or more kinds of fundamental ferroicities coexist, have become one of the hottest topics of condensed matter physics and materials science in recent years. The coexistence of several order parameters in multiferroics brings out novel physical phenomena and offers possibilities for new device functions. The revival of research activities on multiferroics is evidenced by some novel discoveries and concepts, both experimentally and theoretically. In this review, we outline some of the progressive milestones in this stimulating field, especially for those single-phase multiferroics where magnetism and ferroelectricity coexist. First, we highlight the physical concepts of multiferroicity and the current challenges to integrate the magnetism and ferroelectricity into a single-phase system. Subsequently, we summarize various strategies used to combine the two types of order. Special attention is paid to three novel mechanisms for multiferroicity generation: (1) the ferroelectricity induced by the spin orders such as spiral and E-phase antiferromagnetic spin orders, which break the spatial inversion symmetry; (2) the ferroelectricity originating from the charge-ordered states; and (3) the ferrotoroidic system. Then, we address the elementary excitations such as electromagnons, and the application potentials of multiferroics. Finally, open questions and future research opportunities are proposed.

This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.

The study of open quantum systems - microscopic systems exhibiting quantum coherence that are coupled to their environment - has become increasingly important in the past years, as the ability to control quantum coherence on a single particle level has been developed in a wide variety of physical systems. In quantum optics, the study of open systems goes well beyond understanding the breakdown of quantum coherence. There, the coupling to the environment is sufficiently well understood that it can be manipulated to drive the system into desired quantum states, or to project the system onto known states via feedback in quantum measurements. Many mathematical frameworks have been developed to describe such systems, which for atomic, molecular, and optical (AMO) systems generally provide a very accurate description of the open quantum system on a microscopic level. In recent years, AMO systems including cold atomic and molecular gases and trapped ions have been applied heavily to the study of many-body physics, and it has become important to extend previous understanding of open system dynamics in single- and few-body systems to this many-body context. A key formalism that has already proven very useful in this context is the quantum trajectories technique. This method was developed in quantum optics as a numerical tool for studying dynamics in open quantum systems, and falls within a broader framework of continuous measurement theory as a way to understand the dynamics of large classes of open quantum systems. In this article, we review the progress that has been made in studying open many-body systems in the AMO context, focussing on the application of ideas from quantum optics, and on the implementation and applications of quantum trajectories methods in these systems. Control over dissipative processes promises many further tools to prepare interesting and important states in strongly interacting systems, including the realisation of parameter regimes in quantum simulators that are inaccessible via current techniques.

Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet non-trivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard-particle and hard-disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green-Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity κ diverges with system size L as κ ∼ L α . For one-dimensional interacting systems there is strong numerical evidence for a universal exponent α = 1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given.

This article gives an overview of both theoretical and experimental developments concerning states with lattice symmetry breaking in the cuprate high-temperature superconductors. Recent experiments have provided evidence for states with broken rotation as well as translation symmetry, and will be discussed in terms of nematic and stripe physics. Of particular importance here are results obtained using the techniques of neutron and X-ray scattering and scanning tunnelling spectroscopy. Ideas on the origin of lattice-symmetry-broken states will be reviewed, and effective models accounting for various experimentally observed phenomena will be summarized. These include both weak-coupling and strong-coupling approaches, with a discussion of their distinctions and connections. The collected experimental data indicate that the tendency toward uni-directional stripe-like ordering is common to underdoped cuprates, but becomes weaker with increasing number of adjacent CuO 2 layers.

The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in an attempt to delay percolation. This observation of 'explosive percolation' was ultimately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions. Explosive percolation enables many other features such as multiple giant components, modular structures, discrete scale invariance and non-self-averaging, relating to properties found in many real phenomena such as explosive epidemics, electric breakdowns and the emergence of molecular life. Models of explosive synchronization provide an analytic framework for the dynamics of abrupt transitions and reveal the interplay between the distribution in natural frequencies and the network structure, with applications ranging from epileptic seizures to waking from anesthesia. Here we review the vast literature on explosive phenomena in networked systems and synthesize the fundamental connections between models and survey the application areas. We attempt to classify explosive phenomena based on underlying mechanisms and to provide a coherent overview and perspective for future research to address the many vital questions that remained unanswered.

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form 'fluctuations ≈ constant × average α ', where the exponent α is predominantly in the range [1/2, 1]. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names Taylor's law or fluctuation scaling. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling. 1 Dedicated to the memory of L. R. Taylor (1924-2007).

Electrical currents flowing in ferromagnetic materials are spin-polarised as a result of the spin-dependent band structure. When the spatial direction of the polarisation changes, in a domain structure, the electrons must somehow accommodate the necessary change in direction of their spin angular momentum as they pass through the wall. Reflection, scattering, or a transfer of angular momentum onto the lattice are all possible outcomes, depending on the circumstances. This gives rise to a variety of different physical effects, most importantly a contribution to the electrical resistance caused by the wall, and a motion of the wall driven by the spin-polarised current. Historical and recent research on these topics is reviewed.

Relaxor ferroelectrics were discovered in the 1950s but many of their properties are not understood. In this review, we shall concentrate on materials such as PMN (PbMg 1/3 Nb 2/3 O 3 ), which crystallize in the cubic perovskite structure but with the Mg ion, charge 2+, and the Nb ion, charge 5+, randomly distributed over the B site of the perovskite structure. The peak of the dielectric susceptibility for relaxors is much broader in temperature than that of conventional ferroelectrics, while below the maximum of the susceptibility most relaxors remain cubic and show no electric polarization, unlike that observed for conventional ferroelectrics. Because of the large width of the susceptibility, relaxors are often used as capacitors. Recently, there have been many X-ray and neutron scattering studies of relaxors and the results have enabled a more detailed picture to be obtained. An important conclusion is that relaxors can exist in a random field state, as initially proposed by Westphal, Kleemann and Glinchuk, similar to that which has been studied for diluted antiferromagnets. If a relaxor is cooled from a high temperature, then the Burns temperature is a measure of when slow fluctuations become evident. These fluctuations are connected with the disorder and are known as nano-domains. The Burns temperature is not a well-defined transition temperature. At a lower temperature, there is a well-defined boundary to a so-called random field state when the nano-domains become static but there is no long-range periodic order. This phase may have both history-dependent properties and a skin effect in which the surface of the sample is different from that of the bulk material, as also found in experiments on magnetic systems. Section 1 is an introduction to the review, to ferroelectricity and to relaxors. Section 2 gives a description of the results obtained by dielectric, optical, specific heat and other macroscopic properties. These long-wavelength properties give a variety of different characteristic temperatures and do not directly probe the random field state. In Section 3 , we describe the results of neutron and X-ray scattering and show that they strongly support the interpretation that relaxors have a random field state. In Section 4 , we briefly describe the results for other relaxor systems such as (PMN) 1−x (PT) x for which PMN is mixed with different amounts of the ferroelectric lead titanate (PT), and we show that the existence of a random field state enables us also to describe the experimental results for these mixed materials. We hope that this review will inspire further theoretical and experimental work to understand the nature of the random field states and to compare the experimental results more satisfactorily with theory.

This paper reviews the properties and phases of fullerenes and their derivatives and compounds under high pressures. For obvious reasons most of the paper deals with C60 but the materials reviewed also include C70, simple derivatives of C60, carbon nanotubes, and intercalation compounds of C60 with both acceptors and donors, mainly alkali metals. After a brief overview of high-pressure techniques and the structures and properties of C60 at atmospheric pressure, the structural phase diagram of C60 from atmospheric pressure to above 40GPa (400kbar) is reviewed. The evolution with pressure of the orientational and translational structure of 'normal' molecular C60 in the range up to 1-5GPa (depending on temperature) is discussed in some detail, as is the appearance of a large number of polymeric phases at higher pressures and temperatures, some of them known to have extreme mechanical properties. At very high static (or shock) pressures or temperatures, C60 transforms into ordered or disordered forms of diamond or graphite. The phase diagram is reasonably well investigated up to near 10GPa, but at higher pressures there are still large gaps in our knowledge. Available experimental data for the physical properties of both monomeric and polymeric C60 under high pressures are reviewed as far as possible. The compression behaviour of C60 has been well investigated and is discussed in detail because of its basic importance, but optical, electrical and lattice properties have also been studied for several of the many structural phases of C60. Whenever possible, experimental data are compared with the results of theoretical calculations. The phase diagram and properties of C70 are much less known because of the larger complexity caused by the anisotropy of the molecule, and very little is known about most compounds of C60. However, noble-gas intercalation in C60 has been reasonably well investigated. Finally, the high-pressure properties of superconducting alkali-metalintercalated C60 are briefly reviewed.

The physics of ferromagnetic doped manganites, such as La 1-x Ca x MnO 3 with x ≈ 0.2-0.4, is reviewed. The concept of double exchange is discussed within the general framework of itinerant electron magnetism. The new feature in this context is the coupling of electrons to local phonon modes. Emphasis is placed on the quantum nature of the phonons and the link with polaron physics. However it is stressed that the manganites fall into an intermediate coupling regime where standard small-polaron theory does not apply. The recently-developed many-body coherent potential approximation is able to deal with this situation and Green's recent application to the Holstein double-exchange model is described. Issues addressed include the nature of the basic electronic structure, the metal-insulator transition, a unification of colossal magnetoresistance, pressure effects and the isotope effect, pseudogaps in spectroscopy and the effect of electron-phonon coupling on spin waves.

Transparent conductors (TCs) are materials, which are characterized by high transmission of light and simultaneously very high electrical DC conductivity. These materials play a crucial role, and made possible numerous applications in the fields of electro-optics, plasmonics, biosensing, medicine, and "green energy". Modern applications, for example in the field of touchscreen and flexible displays, require that TCs are also mechanically strong and flexible. TC can be broadly classified into two categories: uniform and non-uniform TC. The uniform TC can be viewed as conventional metals (or electron plasmas) with plasma frequency located in the infrared frequency range (e.g. transparent conducting oxides), or ultra-thin metals with large plasma frequency (e.g. graphen). The physics of the nonuniform TC is much more complex, and could involve transmission enhancement due to refraction (including plasmonic), and exotic effects of electron transport, including percolation and fractal effects. This review ties the TC performance to the underlying physical phenomena. We begin with the theoretical basis for studying the various phenomena encountered in TC. Next, we consider the uniform TC, and discuss first the conventional conducting oxides (such as indium tin oxide), reviewing advantages and limitations of these classic uniform electron plasmas. Next, we discuss the potential of single- and multiple-layer graphene as uniform TC. In the part of the paper dealing with non-uniform metallic films, we begin with the review of random metallic networks. The transparency of these networks could be enhanced beyond the classical shading limit by the plasmonic refractive effects. The electrical conduction strongly depends on the network type, and we review first networks made of individual metallic nanowires, where conductivity depends on the inter-wire contact, and the percolation effects. Next, we review the uniform metallic film networks, which are free of the percolation effects and contact problems. In applications that require high-quality electric contact of a TC to an active substrate (such as LED or solar cells), the network performance can be optimized by employing a quasi-fractal structure of the network. We also consider the periodic metallic networks, where active plasmonic refraction leads to the phenomenon of the extraordinary optical transmission. We review the relevant literature on this topic, and demonstrate networks, which take advantage of this strategy (the bio-inspired leaf venation (LV) network, hybrid networks, etc.). Finally, we review "smart" TCs, with an added functionality, such as light interference, metamaterial effects, built-in semiconductors, and their junctions.

Although nineteen years have passed since the discovery of high temperature cuprate superconductivity 1 , there is still no consensus on its physical origin. This is in large part because of a lack of understanding of the state of matter out of which the superconductivity arises. In optimally and underdoped materials, this state exhibits a pseudogap at temperatures large compared to the superconducting transition temperature 2 , 3 . Although discovered only three years after the pioneering work of Bednorz and Müller, the physical origin of this pseudogap behavior and whether it constitutes a distinct phase of matter is still shrouded in mystery. In the summer of 2004, a band of physicists gathered for five weeks at the Aspen Center for Physics to discuss the pseudogap. In this perspective, we would like to summarize some of the results presented there and discuss the importance of the pseudogap phase in the context of strongly correlated electron systems.

The Kondo lattice model introduced in 1977 describes a lattice of localized magnetic moments interacting with a sea of conduction electrons. It is one of the most important canonical models in the study of a class of rare earth compounds, called heavy fermion systems, and as such has been studied intensively by a wide variety of techniques for more than a quarter of a century. This review focuses on the one-dimensional case at partial band filling, in which the number of conduction electrons is less than the number of localized moments. The theoretical understanding, based on the bosonized solution, of the conventional Kondo lattice model is presented in great detail. This review divides naturally into two parts, the first relating to the description of the formalism, and the second to its application. After an all-inclusive description of the bosonization technique, the bosonized form of the Kondo lattice Hamiltonian is constructed in detail. Next the double-exchange ordering, Kondo singlet formation, the RKKY interaction and spin polaron formation are described comprehensively. An in-depth analysis of the phase diagram follows, with special emphasis on the destruction of the ferromagnetic phase by spin-flip disorder scattering, and of recent numerical results. The results are shown to hold for both antiferromagnetic and ferromagnetic Kondo lattice. The general exposition is pedagogic in tone.

We review developments in the theoretical description and understanding of plutonium in terms of a metal with itinerant (band) 5f electrons. Within this picture most facets of this remarkable and anomalous material are accurately described by first-principle, parameter-free, density-functional-theory (DFT) calculations. We show that the model explains plutonium's phase stability, elasticity, lattice vibrations, electronic structure, alloy properties, and magnetism. Fluctuations are addressed by means of constrained DFT calculations and new light is shed on the anomalous properties of δ plutonium, including explaining its negative thermal expansion. Effects of alloying and point defects in plutonium are also addressed. It is further emphasized that strong electron correlations, originating from a large intra-atomic Coulomb repulsion (∼4 eV) of the 5f electrons, that has often been assumed for plutonium in the literature, is inconsistent with the experimental phase diagram of plutonium.

Recent advances in understanding model systems of wet granular materials are presented, with particular emphasis on statistical concepts, dynamics, and phase transitions. It is demonstrated that although wet granular systems are quite complex, their main features may be understood on the basis of rather simple concepts. The significance of these systems for investigating fundamental problems of non-equilibrium dynamics are shortly discussed.

This review focuses on recent developments in the theoretical, numerical and experimental study of slow dynamics in colloidal systems, with a particular emphasis on the glass transition phenomenon. Colloidal systems appear to be particularly suited for tackling the general problem of dynamic arrest, since they show a larger flexibility compared to atomic and molecular glasses because of their size and the possibility of manipulating the physical and chemical properties of the samples. Indeed, a wealth of new effects, not easily observable in molecular liquids, have been predicted and measured in colloidal systems. The slow dynamic behavior of three classes of colloidal suspension is reviewed - hard colloids, short-range attractive colloids and soft colloidal systems - selecting the model systems among the most prominent candidates for grasping the essential features of dynamic arrest. Emphasis is on the possibility of performing a detailed comparison between experimental data and theoretical predictions based on the mode coupling theory of the glass transition. Finally, the importance of understanding the system's kinetic arrest phase diagram, i.e. the regions in phase space where disordered arrested states can be expected, is stressed. When and how these states are kinetically stabilized with respect to the ordered lowest free energy phases is then examined in order to provide a framework for interpreting and developing new ideas in the study of new materials. Contents PAGE 1. Introduction 472 2. Theoretical background 473 2.1. Mode coupling theory 473 2.2. MCT features 475 2.3. Other theoretical approaches 479 3. Hard colloids 480 3.1. Hard sphere colloids 481 3.2. Polydisperse hard spheres 485 3.3. Hard ellipsoids 487 4. Attractive colloids 490 4.1. MCT predictions 491 4.2. Experiments 494 4.3. Numerical studies 498 4.4. Higher-order singularities 499 4.5. Mechanical properties 500 4.6. Remarks on attractive colloids 502 5. Soft colloids 504 5.1. Charged colloids and the Wigner glass 504 5.2. Competing interactions: cluster phases 509 5.3. Ultrasoft colloids: star polymers 515 6. Perspectives and conclusions 517 Acknowledgements 518 References 518

Ca 2+ is one of the most important messengers. It transmits signals inside living cells and takes part in intercellular coordination. The dynamics of the Ca 2+ concentration shows a transition from elemental, stochastic events to global events like waves and oscillations. This transition renders it an ideal tool for studying basic concepts of pattern formation, especially since access to the most important experimental parameters is given. Ca 2+ dynamics in living cells has been a major topic of biophysical modelling in the last 15 years. Modelling has reached the level of predictive power. The theoretical analysis of waves provided new insight into the mechanisms of Ca 2+ signaling and led to new concepts of analysis of wave equations with concentration dependent diffusion and novel wave bifurcations. Modelling of oscillations provided understanding especially of complex oscillations and allowed to extract information about the underlying cellular parameters and mechanisms. The investigation of the stochastic aspects of intracellular Ca 2+ dynamics demonstrated the fundamental role of fluctuations arising from the control of the release channel by Ca 2+ and IP 3 . This review presents an overview of current theoretical research on Ca 2+ dynamics in living cells driven by the inositol 1,4,5-trisphosphate receptor channel.