Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory

Strong electronic correlations pose one of the biggest challenges to solid state theory. We review recently developed methods that address this problem by starting with the local, eminently important correlations of dynamical mean field theory (DMFT). On top of this, non-local 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. We provide an overview of the successes and results achieved---hitherto mainly for model Hamiltonians---and outline future prospects for realistic material calculations.Comment: 60 pages, 42 figures, replaced by the version to be published in Rev. Mod. Phys. 201

Beyond extended dynamical mean-field theory: Dual boson approach to the two-dimensional extended Hubbard model

The dual boson approach [Ann. Phys. 327, 1320 (2012)] provides a means to construct a diagrammatic expansion around the extended dynamical mean-field theory (EDMFT). In this paper, we present the numerical implementation of the approach and apply it to the extended Hubbard model with nearest-neighbor interaction $V$. We calculate the EDMFT phase diagram and study the effect of diagrams beyond EDMFT on the transition to the charge-ordered phase. Including diagrammatic corrections to the EDMFT polarization shifts the EDMFT phase boundary to lower values of $V$. The approach interpolates between the random phase approximation in the weak coupling limit and EDMFT for strong coupling. Neglecting vertex corrections leads to results reminiscent of the EDMFT+$GW$ approximation. We however find significant deviations from the dual boson results already for small values of the interaction, emphasizing the crucial importance of fermion-boson vertex corrections.Comment: Published version; 24 pages, 32 figure

A Quantum Monte Carlo algorithm for non-local corrections to the Dynamical Mean-Field Approximation

We present the algorithmic details of the dynamical cluster approximation (DCA), with a quantum Monte Carlo (QMC) method used to solve the effective cluster problem. The DCA is a fully-causal approach which systematically restores non-local correlations to the dynamical mean field approximation (DMFA) while preserving the lattice symmetries. The DCA becomes exact for an infinite cluster size, while reducing to the DMFA for a cluster size of unity. We present a generalization of the Hirsch-Fye QMC algorithm for the solution of the embedded cluster problem. We use the two-dimensional Hubbard model to illustrate the performance of the DCA technique. At half-filling, we show that the DCA drives the spurious finite-temperature antiferromagnetic transition found in the DMFA slowly towards zero temperature as the cluster size increases, in conformity with the Mermin-Wagner theorem. Moreover, we find that there is a finite temperature metal to insulator transition which persists into the weak-coupling regime. This suggests that the magnetism of the model is Heisenberg like for all non-zero interactions. Away from half-filling, we find that the sign problem that arises in QMC simulations is significantly less severe in the context of DCA. Hence, we were able to obtain good statistics for small clusters. For these clusters, the DCA results show evidence of non-Fermi liquid behavior and superconductivity near half-filling.Comment: 25 pages, 15 figure

Merging GW with DMFT and non-local correlations beyond

We review recent developments in electronic structure calculations that go beyond state-of-the-art methods such as density functional theory (DFT) and dynamical mean field theory (DMFT). Specifically, we discuss the following methods: GW as implemented in the Vienna {\it ab initio} simulation package (VASP) with the self energy on the imaginary frequency axis, GW+DMFT, and ab initio dynamical vertex approximation (D$\Gamma$A). The latter includes the physics of GW, DMFT and non-local correlations beyond, and allows for calculating (quantum) critical exponents. We present results obtained by the three methods with a focus on the benchmark material SrVO$_3$.Comment: tutorial review submitted to EPJ-ST (scientific report of research unit FOR 1346); 11 figures 27 page

Renormalization group approaches to strongly correlated electron systems

Strongly correlated electron systems host a plethora of fascinating physical phenomena and pose formidable challenges in their theoretical analysis. The challenges originate from the inherent complexity of the quantum many-body problem - no classical computer will ever be able to fully simulate these systems - and the lack of an effective single-particle picture, as the strong mutual interactions make it impossible to regard the electrons as independent from each other. As a consequence, most systems of correlated electrons can only be tackled approximately and numerically. In this thesis, we develop a set of numerical methods for strongly correlated electrons, which are inspired by the renormalization group (RG) idea of including degrees of freedom successively from high to low energies. This enables an efficient organization of the diverse fluctuations and is key for an accurate treatment of interacting quantum systems, where collective behavior and composite objects emerge at energy scales far below those of the microscopic constituents. In a first part, we consider the functional renormalization group (fRG), a versatile framework to study the flow of correlation functions upon modulating the underlying action. Though widely used, it has often acted more as a qualitative rather than quantitative method, due a nontransparent approximation induced by truncating the hierarchy of flow equations. We develop an iterative multiloop fRG (mfRG) scheme, which ameliorates this approximation and eliminates many of the drawbacks of fRG experienced hitherto. In particular, it restores the independence of results on the choice of RG regulator and establishes a rigorous relation to the parquet formalism. Furthermore, we show how to derive the flow equations directly from self-consistent many-body relations. This establishes a form of diagrammatic resummations at the two-particle level which circumvents ill-behaved two-particle-irreducible vertices. An application to the prototypical two-dimensional Hubbard model illustrates how our multiloop scheme elevates the fRG approach to correlated electron systems to a quantitative level. Secondly, we employ the numerical renormalization group (NRG), based on the iterative diagonalization of impurity Hamiltonians, in conjunction with the dynamical mean-field theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, real-frequency impurity solver for DMFT. It has been pivotal to the understanding of Hund metals, where strong correlations arise from Hund's rules even at moderate Coulomb repulsion. Building on recent methodological advances, we extend the range of application of DMFT+NRG from orbital-degenerate models to more realistic setups: We first study orbital differentiation in a three-orbital Hund-metal model and unravel key effects of the orbital-selective Mott transition. In a real-materials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hund-metal material Sr2RuO4. We particularly follow its RG flow to the Fermi-liquid regime at previously inaccessible low temperatures and generally present DFT+DMFT+NRG as a new computational paradigm for strongly correlated materials. As a side project of our fRG work, we develop an algorithm to count Feynman diagrams from closed many-body relations, which reveals the surprising outcome that totally irreducible contributions are responsible for the factorial growth in the number of diagrams. Additionally, we use NRG to study transport through three-level quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium

Renormalization group approaches to strongly correlated electron systems

Strongly correlated electron systems host a plethora of fascinating physical phenomena and pose formidable challenges in their theoretical analysis. The challenges originate from the inherent complexity of the quantum many-body problem - no classical computer will ever be able to fully simulate these systems - and the lack of an effective single-particle picture, as the strong mutual interactions make it impossible to regard the electrons as independent from each other. As a consequence, most systems of correlated electrons can only be tackled approximately and numerically. In this thesis, we develop a set of numerical methods for strongly correlated electrons, which are inspired by the renormalization group (RG) idea of including degrees of freedom successively from high to low energies. This enables an efficient organization of the diverse fluctuations and is key for an accurate treatment of interacting quantum systems, where collective behavior and composite objects emerge at energy scales far below those of the microscopic constituents. In a first part, we consider the functional renormalization group (fRG), a versatile framework to study the flow of correlation functions upon modulating the underlying action. Though widely used, it has often acted more as a qualitative rather than quantitative method, due a nontransparent approximation induced by truncating the hierarchy of flow equations. We develop an iterative multiloop fRG (mfRG) scheme, which ameliorates this approximation and eliminates many of the drawbacks of fRG experienced hitherto. In particular, it restores the independence of results on the choice of RG regulator and establishes a rigorous relation to the parquet formalism. Furthermore, we show how to derive the flow equations directly from self-consistent many-body relations. This establishes a form of diagrammatic resummations at the two-particle level which circumvents ill-behaved two-particle-irreducible vertices. An application to the prototypical two-dimensional Hubbard model illustrates how our multiloop scheme elevates the fRG approach to correlated electron systems to a quantitative level. Secondly, we employ the numerical renormalization group (NRG), based on the iterative diagonalization of impurity Hamiltonians, in conjunction with the dynamical mean-field theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, real-frequency impurity solver for DMFT. It has been pivotal to the understanding of Hund metals, where strong correlations arise from Hund's rules even at moderate Coulomb repulsion. Building on recent methodological advances, we extend the range of application of DMFT+NRG from orbital-degenerate models to more realistic setups: We first study orbital differentiation in a three-orbital Hund-metal model and unravel key effects of the orbital-selective Mott transition. In a real-materials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hund-metal material Sr2RuO4. We particularly follow its RG flow to the Fermi-liquid regime at previously inaccessible low temperatures and generally present DFT+DMFT+NRG as a new computational paradigm for strongly correlated materials. As a side project of our fRG work, we develop an algorithm to count Feynman diagrams from closed many-body relations, which reveals the surprising outcome that totally irreducible contributions are responsible for the factorial growth in the number of diagrams. Additionally, we use NRG to study transport through three-level quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium

The ALF (Algorithms for Lattice Fermions) project release 2.0. Documentation for the auxiliary-field quantum Monte Carlo code

The Algorithms for Lattice Fermions package provides a general code for the finite-temperature and projective auxiliary-field quantum Monte Carlo algorithm. The code is engineered to be able to simulate any model that can be written in terms of sums of single-body operators, of squares of single-body operators and single-body operators coupled to a bosonic field with given dynamics. The package includes five pre-defined model classes: SU(N) Kondo, SU(N) Hubbard, SU(N) t-V and SU(N) models with long range Coulomb repulsion on honeycomb, square and N-leg lattices, as well as $Z_2$ unconstrained lattice gauge theories coupled to fermionic and $Z_2$ matter. An implementation of the stochastic Maximum Entropy method is also provided. One can download the code from our Git instance at https://git.physik.uni-wuerzburg.de/ALF/ALF/-/tree/ALF-2.0 and sign in to file issues.Comment: 121 pages, 11 figures. v3: quick tutorial section added, typos corrected, etc. Submission to SciPost. arXiv admin note: text overlap with arXiv:1704.0013

Molecular Dynamics Simulation

Condensed matter systems, ranging from simple fluids and solids to complex multicomponent materials and even biological matter, are governed by well understood laws of physics, within the formal theoretical framework of quantum theory and statistical mechanics. On the relevant scales of length and time, the appropriate ‘first-principles’ description needs only the Schroedinger equation together with Gibbs averaging over the relevant statistical ensemble. However, this program cannot be carried out straightforwardly—dealing with electron correlations is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the many-body problem can be effectively reduced to one of independent particles or quasi-particles. [...