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 numerically exact continuous-time Quantum Monte Carlo algorithm for fermions with a general interaction non-local in space-time. The new determinantal grand-canonical scheme is based on a stochastic series expansion for the partition function in the interaction representation. The method is particularly applicable for multi-band, time-dependent correlations since it does not invoke the Hubbard-Stratonovich transformation. The test calculations for exactly solvable models, as well results for the Green function and for the time-dependent susceptibility of the multi-band super-symmetric model with a spin-flip interaction are discussed.

A new diagrammatic technique is developed to describe nonlocal effects (e.g., pseudogap formation) in the Hubbard-like models. In contrast to cluster approaches, this method utilizes an exact transition to the dual set of variables, and it therefore becomes possible to treat the irreducible vertices of an effective single-impurity problem as small parameters. This provides a very efficient interpolation between weak-coupling (band) and atomic limits. The antiferromagnetic pseudogap formation in the Hubbard model is correctly reproduced by just the lowest-order diagrams. Unfortunately, the analytical treatment of these problems is essentially restricted by the lack of explicit small parameters for the most physically interesting interactions. Direct numerical methods, such as exact diagonalization [7] or quantum Monte Carlo (QMC) [8,9] are limited by the clusters being of a relatively small size, or face other obstacles such as the famous sign problem for QMC simulations at low temperature [10]. There is a very successful approximate way to treat these models via the framework of so-called dynamical mean-field theory (DMFT) [5], where the lattice many-body problem is replaced with an effective impurity model. This approach is essentially based on the assumption of a local (i.e. momentum-independent) fermionic self energy. Indeed, there are numerous interesting phenomena which are basically determined by local electron correlations, such as Kondo effect [11], Mott-Hubbard transitions [5] and local moment formation in itinerant-electron magnets [12]. At the same time, momentum dependence of the self energy is of crucial importance for Luttinger liquid formation in low-dimensional systems [3,13], d-wave pairing in high-T c superconductors [2,14,15], and nonFermi-liquid behavior due to van Hove singularities in two dimensions [16]. Recently a rather strong momentum dependence of the effective mass renormalization in photoemission spectra of iron was observed [17].Currently, non-local many body effects in strongly correlated systems are mainly studied via the framework of various cluster generalizations of DMFT [14,15,18,19]. Cluster methods do catch basic physics of d-wave pairing and antiferromagnetism in high-T c superconductors [14,15], and the effects of intercite Coulomb interaction in various transition-metal oxides [20,21,22]. At the same time, however effects like Luttinger liquid formation or van Hove singularities can not be described in cluster approaches. In such cases the correlations are essentially long-ranged and it is more natural to describe the correlations in momentum space. Recently attempts have been made to consider non local correlation effects in momentum space starting from DMFT as a zeroth-order approximation [23,24]. This approach requires a solution of ladder-like integral equation for complete vertex Γ and the subsequent use of the Bethe-Salpeter equation to obtain Green's functions. The first step here exploits an irreducible vertex of the effective impurity problem γ (4) , whereas the ...

Strong electronic correlations pose one of the biggest challenges to solid state theory. Recently developed methods that address this problem by starting with the local, eminently important correlations of dynamical mean field theory (DMFT) are reviewed. In addition, nonlocal correlations on all length scales are generated through Feynman diagrams, with a local two-particle vertex instead of the bare Coulomb interaction as a building block. With these diagrammatic extensions of DMFT long-range charge-, magnetic-, and superconducting fluctuations as well as (quantum) criticality can be addressed in strongly correlated electron systems. An overview is provided of the successes and results achieved mainly for model Hamiltonians and an outline is given of future prospects for realistic material calculations. PACS numbers: 71.10.-w,71.10.Fd,71.27.+a CONTENTS 42 5. One and zero dimensions 43 B. Heavy fermions and Kondo lattice model (KLM) 44 C. Falicov-Kimball (FK) model 45 D. Models of Disorder 47 E. Non-local interactions and multiorbitals 48 V. Open source implementations 51 VI. Conclusion and outlook 51 References 53 arXiv:1705.00024v2 [cond-mat.str-el]