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]

Electronic correlated systems are often well described by dynamical mean field theory (DMFT). While DMFT studies have mainly focused hitherto on one-particle properties, valuable information is also enclosed into local two-particle Green's functions and vertices. They represent the main ingredient to compute momentum-dependent response functions at the DMFT level and to treat non-local spatial correlations at all length scales by means of diagrammatic extensions of DMFT. The aim of this paper is to present a DMFT analysis of the local reducible and irreducible two-particle vertex functions for the Hubbard model in the context of an unified diagrammatic formalism. An interpretation of the observed frequency structures is also given in terms of perturbation theory, of the comparison with the atomic limit, and of the mapping onto the attractive Hubbard model.

We have studied the impact of non-local electronic correlations at all length scales on the MottHubbard metal-insulator transition in the unfrustrated two-dimensional Hubbard model. Combining dynamical vertex approximation, lattice quantum Monte-Carlo and variational cluster approximation, we demonstrate that scattering at long-range fluctuations, i.e., Slater-like paramagnons, opens a spectral gap at weak-to-intermediate coupling -irrespectively of the preformation of localized or short-ranged magnetic moments. This is the reason, why the two-dimensional Hubbard model has a paramagnetic phase which is insulating at low enough temperatures for any (finite) interaction and no Mott-Hubbard transition is observed. Introduction.The Mott-Hubbard metal-insulator transition (MIT) [1] is one of the most fundamental hallmarks of the physics of electronic correlations. Nonetheless, astonishingly little is known exactly, even for its simplest modeling, i.e., the single-band Hubbard Hamiltonian [2]: Exact solutions for this model are available only in the extreme, limiting cases of one and infinite dimensions.In one dimension (1D), the Bethe ansatz shows that there is actually no Mott-Hubbard transition [3][4][5]; or, in other words, it occurs for a vanishingly small Hubbard interaction U : At any U > 0 the 1D-Hubbard model is insulating at half filling. One dimension is, however, rather peculiar: While there is no antiferromagnetic ordering even at temperature T = 0, antiferromagnetic spin fluctuations are strong and long-ranged, decaying slowly, i.e., algebraically. Also the (doped) metallic phase is not a standard Fermi liquid but a Luttinger liquid.For the opposite extreme, infinite dimensions, the dynamical mean field theory (DMFT) [6] becomes exact [7], which allows for a clear-cut and -to a certain extentalmost "idealized" description of a pure Mott-Hubbard MIT. In fact, since in D = ∞ only local correlations survive [7], the Mott-Hubbard insulator of DMFT consists of a collection of localized (but not long-range ordered) magnetic moments. This way, if antiferromagnetic order is neglected or sufficiently suppressed, DMFT describes a first-order MIT [6,8], ending with a critical endpoint.As an approximation, DMFT is applicable to the more realistic cases of the three-and two-dimensional Hubbard models. However, the DMFT description of the MIT is the very same here, since only the non-interacting density of states (DOS) and in particular its second moment enter. This is a natural shortcoming of the mean-field nature of DMFT: antiferromagnetic fluctuations have no effect at all on the DMFT spectral function or self-energy above the antiferromagnetic ordering temperature T N .In 3D, antiferromagnetic fluctuations reduce T N siz-

Identifying the fingerprints of the Mott-Hubbard metal-insulator transition may be quite elusive in correlated metallic systems if the analysis is limited to the single particle level. However, our dynamical mean-field calculations demonstrate that the situation changes completely if the frequency dependence of the two-particle vertex functions is considered: The first nonperturbative precursors of the Mott physics are unambiguously identified well inside the metallic regime by the divergence of the local Bethe-Salpeter equation in the charge channel. In the low-temperature limit this occurs for interaction values where incoherent high-energy features emerge in the spectral function, while at high temperatures it is traceable up to the atomic limit.

We demonstrate how to identify which physical processes dominate the low-energy spectral functions of correlated electron systems. We obtain an unambiguous classification through an analysis of the equation of motion for the electron self-energy in its charge, spin, and particle-particle representations. Our procedure is then employed to clarify the controversial physics responsible for the appearance of the pseudogap in correlated systems. We illustrate our method by examining the attractive and repulsive Hubbard model in two dimensions. In the latter, spin fluctuations are identified as the origin of the pseudogap, and we also explain why d-wave pairing fluctuations play a marginal role in suppressing the low-energy spectral weight, independent of their actual strength. Introduction.-Correlated electron systems display some of the most fascinating phenomena in condensed matter physics, but their understanding still represents a formidable challenge for theory and experiments. For photoemission [1] or STM [2,3] spectra, which measure single-particle quantities, information about correlation is encoded in the electronic self-energy Σ. However, due to the intrinsically many-body nature of the problems, even an exact knowledge of Σ is not sufficient for an unambiguous identification of the underlying physics. A perfect example of this is the pseudogap observed in the single-particle spectral functions of underdoped cuprates [4], and, more recently, of their nickelate analogues [5]. Although relying on different assumptions, many theoretical approaches provide self-energy results compatible with the experimental spectra. This explains the lack of a consensus about the physical origin of the pseudogap: In the case of cuprates, the pseudogap has been attributed to spin fluctuations [6-10], preformed pairs [11][12][13][14][15], Mottness [16,17], and, recently, to the interplay with charge fluctuations [18][19][20][21] or to Fermi-liquid scenarios [22]. The existence and the role of (d-wave) superconducting fluctuations [11][12][13][14][15] in the pseudogap regime are still openly debated for the basic model of correlated electrons, the Hubbard model.Experimentally, the clarification of many-body physics is augmented by a simultaneous investigation at the two-particle level, i.e., via neutron scattering [23], infrared or optical [24] and pump-probe spectroscopy [25], muon-spin relaxation [26], and correlation or coincidence two-particle spectroscopies [27][28][29]. Analogously, theoretical studies of Σ can also be supplemented by a corresponding analysis at the two-particle level. In this

We present a novel scheme for an unbiased, nonperturbative treatment of strongly correlated fermions. The proposed approach combines two of the most successful many-body methods, the dynamical mean field theory and the functional renormalization group. Physically, this allows for a systematic inclusion of nonlocal correlations via the functional renormalization group flow equations, after the local correlations are taken into account nonperturbatively by the dynamical mean field theory. To demonstrate the feasibility of the approach, we present numerical results for the two-dimensional Hubbard model at half filling.

By means of the dynamical vertex approximation (DΓA) we include spatial correlations on all length scales beyond the dynamical mean field theory (DMFT) for the half-filled Hubbard model in three dimensions. The most relevant changes due to non-local fluctuations are: (i) a deviation from the mean-field critical behavior with the same critical exponents as for the three dimensional Heisenberg (anti)-ferromagnet and (ii) a sizable reduction of the Néel temperature (TN ) by ∼ 30% for the onset of antiferromagnetic order. Finally, we give a quantitative estimate of the deviation of the spectra between DΓA and DMFT in different regions of the phase-diagram.PACS numbers: 71.10. Fd, 71.27.+a Almost 50 years after the invention of the Hubbard model [1] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [2-4] was a big step forward to calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e, the local one, is well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in purely electronic models [5]. However, nonlocal spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of second-order phase transitions. As these correlations are neglected in DMFT, this scheme provides only for a conventional mean-field (MF) description of the critical properties.To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed [6]. In these approaches spatial correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of d = 3-clusters is restricted to about 100 sites. Hence, short-range correlations are included by these approaches, whereas long-range ones are not (e.g. for spacings larger than 5 lattice sites in d = 3). Nonetheless, Kent et al. [7] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the Néel temperature of the paramagneticto-antiferromagnetic phase transition which was found in agreement with earlier lattice quantum Monte Carlo (QMC) results by Staudt et al. [8].As an alternative to cluster extensions and, in particular, to include long-range correlations on an equal footing, more recently diagrammatic expansions of DMFT have been proposed: (i) the DMFT plus spin-fermion model [9], (ii) the dynamical vertex approximation (DΓA) [10][11][12] which approximates the fully irreducible n-particle vertex to be local [10] or that of a DCA cluster [12...

We present an approach which is based on the one-particle irreducible (1PI) generating functional formalism and includes electronic correlations on all length scales beyond the local correlations of dynamical mean-field theory (DMFT). This formalism allows us to unify aspects of the dynamical vertex approximation (D A) and the dual fermion (DF) scheme, yielding a consistent formulation of nonlocal correlations at the one-and two-particle level beyond DMFT within the functional integral formalism. In particular, the considered approach includes one-particle reducible contributions from the three-and more-particle vertices in the dual fermion approach, as well as some diagrams not included in the ladder version of D A. To demonstrate the applicability and physical content of the 1PI approach, we compare the diagrammatics of 1PI, DF, and D A, as well as the numerical results of these approaches for the half-filled Hubbard model in two dimensions.