Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of "correlated electron" materials as auxiliary problems whose solution gives the "dynamical mean field" approximation to the self energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. We present derivations and descriptions of the algorithms in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied and outline prospects for future applications. 16 1. Measurement of the Green's function 16 2. Role of the parameter K -potential energy 16 V. Hybridization expansion solvers CT-HYB 16 A. The hybridization expansion representation 16 B. Density -density interactions 17 C. Formulation for general interactions

We present a new continuous-time solver for quantum impurity models such as those relevant to dynamical mean field theory. It is based on a stochastic sampling of a perturbation expansion in the impurity-bath hybridization parameter. Comparisons with Monte Carlo and exact diagonalization calculations confirm the accuracy of the new approach, which allows very efficient simulations even at low temperatures and for strong interactions. As examples of the power of the method we present results for the temperature dependence of the kinetic energy and the free energy, enabling an accurate location of the temperature-driven metal-insulator transition.

We present a comprehensive theoretical treatment of the effect of electron-phonon interactions on molecular transistors, including both quantal and classical limits. We study both equilibrated and out of equilibrium phonons. We present detailed results for conductance, noise and phonon distribution in two regimes. One involves temperatures large as compared to the rate of electronic transitions on and off the dot; in this limit our approach yields classical rate equations, which are solved numerically for a wide range of parameters. The other regime is that of low temperatures and weak electron-phonon coupling where a perturbative approximation in the Keldysh formulation can be applied. The interplay between the phonon-induced renormalization of the density of states on the quantum dot and the phonon-induced renormalization of the dot-lead coupling is found to be important. Whether or not the phonons are able to equilibrate in a time rapid compared to the transit time of an electron through the dot is found to affect the conductance. Observable signatures of phonon equilibration are presented. We also discuss the nature of the low-T to high-T crossover.

The high-temperature copper oxide superconductors are of fundamental and enduring interest. They not only manifest superconducting transition temperatures inconceivable 15 years ago, but also exhibit many other properties apparently incompatible with conventional metal physics. The materials expand our notions of what is possible, and compel us to develop new experimental techniques and theoretical concepts. This article provides a perspective on recent developments and their implications for our understanding of interacting electrons in metals.

We use analytic techniques and the dynamical mean field method to study the crossover from fermi liquid to polaron behavior in models of electrons interacting with dispersionless classical phonons.

The discovery of spectacularly large magnetoresistive responses in a class of metallic manganese oxides has raised hopes that these compounds might be of practical utility. But regardless of whether this promise is realized, these materials provide an ideal system in which to elucidate the properties of metals in which electron-lattice interactions play a key role.Transition-metal oxides have long been the subject of study, because they exhibit a wide range of exotic and still imperfectly understood structural, magnetic and electronic behaviour. This behaviour cannot be explained within the context of the usual one-electron band theory that accounts well for the properties of most other solids, indicating the importance of strong electron-electron and electron-lattice correlations. The properties of these materials continue to surprise; for example, 1986 saw the discovery of hightemperature superconductivity in materials based on copper oxide.More recently, attention has become focused on a certain class of manganese oxides, the manganite perovskites. Although these materials have been studied for many years 1 , the current burst of activity was stimulated by reports by Helmholt et al. 2 and Jin et al. 3 of spectacularly large-''colossal'' as Jin et al. 3 put it-magnetoresistance in this family of compounds. Magnetoresistance, the variation of electrical resistance with magnetic field, is crucial to several areas of technology, such as magnetic data storage, and much of the impetus for the present interest in the manganites stems from the possible utility of their magnetoresistive properties. Whether the manganite perovskites currently under consideration will prove technologically useful is still far from clear. But the observation of colossal magnetoresistance has stimulated a considerable amount of work aimed mainly at understanding and improving their magnetoresistive properties, and at examining other related classes of transition-metal oxides (such as spinels, pyrochlores and magnetites) which display similar behaviour and may have more technologically convenient properties.But here I will argue that the manganites are also important for basic condensed-matter physics for quite a different reason. In these materials, the interaction between the electrons and lattice vibrations (phonons) is unusually strong, leading to a wide range of striking physical phenomena and, most crucially, can be 'tuned' over a wide range by variation of chemical composition, temperature and magnetic field. These materials therefore provide an unprecedented opportunity to study the poorly understood physics of systems in which a high density of electrons is strongly coupled to phonons and, in particular, to elucidate the interplay between local structural deformations and global properties. This interplay is becoming the focus of attention in many contexts, including conducting polymers 4 and ferroelectrics 5 (for a summary of other recent work on local structures, see ref. 6).

The concept of quantum criticality is proving to be central to attempts to understand the physics of strongly correlated electrons. Here, we argue that observations on the itinerant metamagnet Sr3Ru2O7 represent good evidence for a new class of quantum critical point, arising when the critical end point terminating a line of first-order transitions is depressed toward zero temperature. This is of interest both in its own right and because of the convenience of having a quantum critical point for which the tuning parameter is the magnetic field. The relationship between the resultant critical fluctuations and novel behavior very near the critical field is discussed.

Surface science is an important and well-established branch of materials science involving the study of changes in material properties near a surface or interface. A fundamental issue has been atomic reconstruction: how the surface lattice symmetry differs from the bulk. 'Correlated-electron compounds' are materials in which strong electron-electron and electron-lattice interactions produce new electronic phases, including interaction-induced (Mott) insulators, many forms of spin, charge and orbital ordering, and (presumably) high-transition-temperature superconductivity. Here we propose that the fundamental issue for the new field of correlated-electron surface/interface science is 'electronic reconstruction': how does the surface/interface electronic phase differ from that in the bulk? As a step towards a general understanding of such phenomena, we present a theoretical study of an interface between a strongly correlated Mott insulator and a band insulator. We find dramatic interface-induced electronic reconstructions: in wide parameter ranges, the near-interface region is metallic and ferromagnetic, whereas the bulk phase on either side is insulating and antiferromagnetic. Extending the analysis to a wider range of interfaces and surfaces is a fundamental scientific challenge and may lead to new applications for correlated electron materials.