arXiv subjects

G. Rohringer

Publications and source records attributed to G. Rohringer.

At least 19 recordsLinked to original sources

arXiv2020

High-frequency asymptotics of the vertex function: diagrammatic parametrization and algorithmic implementation

Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet-equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and non-local static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.

cond-mat.str-el↗

arXiv2019

Dual parquet scheme for the two-dimensional Hubbard model: modelling low-energy physics of high-$T_c$ cuprates with high momentum resolution

We present a new method to treat the two-dimensional (2D) Hubbard model for parameter regimes which are relevant for the physics of the high-$T_c$ superconducting cuprates. Unlike previous attempts to attack this problem, our new approach takes into account all fluctuations in different channels on equal footing and is able to treat reasonable large lattice sizes up to 32x32. This is achieved by the following three-step procedure: (i) We transform the original problem to a new representation (dual fermions) in which all purely local correlation effects from the dynamical mean field theory are already considered in the bare propagator and bare interaction of the new problem. (ii) The strong $1/(iν)^2$ decay of the bare propagator allows us to integrate out all higher Matsubara frequencies besides the lowest using low order diagrams. The new effective action depends only on the two lowest Matsubara frequencies which allows us to, (iii) apply the two-particle self-consistent parquet formalism, which takes into account the competition between different low-energy bosonic modes in an unbiased way, on much finer momentum grids than usual. In this way, we were able to map out the phase diagram of the 2D Hubbard model as a function of temperature and doping. Consistently with the experimental evidence for hole-doped cuprates and previous dynamical cluster approximation calculations, we find an antiferromagnetic region at low-doping and a superconducting dome at higher doping. Our results also support the role of the van Hove singularity as an important ingredient for the high value of $T_c$ at optimal doping. At small doping, the destruction of antiferromagnetism is accompanied by an increase of charge fluctuations supporting the scenario of a phase separated state driven by quantum critical fluctuations.

cond-mat.str-el↗

arXiv2019

Fate of a topological invariant for correlated lattice electrons at finite temperature

Electrons on a two-dimensional (2$d$) lattice which is exposed to a strong uniform magnetic field show intriguing physical phenomena. The spectrum of such systems exhibits a complex (multi-)band structure known as Hofstadter's butterfly. For fillings at which the system is a band insulator one observes a quantized integer-valued Hall conductivity $σ_{xy}$ corresponding to a topological invariant, the first Chern number $\mathcal{C}_1$. This is robust against many-body interactions as long as no changes in the gap structure occur. Strictly speaking, this stability holds only at zero temperatures $T$ while for $T>0$ correlation effects have to be taken into account. In this work, we address this question by presenting a dynamical mean field theory (DMFT) study of the Hubbard model in a uniform magnetic field. The inclusion of local correlations at finite temperature leads to (i) a shrinking of the integer plateaus of $σ_{xy}$ as a function of the chemical potential and (ii) eventually to a deviation from these integer values. We demonstrate that these effects can be related to a correlation-driven narrowing and filling of the band gap, respectively.

cond-mat.str-el↗

arXiv2018

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.

cond-mat.str-el↗

arXiv2017

Role of three-particle vertex within dual fermion calculations

We investigate the influence of self-energy diagrams beyond the two-particle vertex level within dual fermion theory. Specifically, we calculate the local three-particle vertex and construct from it selected dual fermion self-energy corrections to dynamical mean field theory. For the two-dimensional Hubbard model, the thus obtained self-energy corrections are small in the parameter space where dual fermion corrections based on the two-particle vertex only are small. However, in other parts of the parameter space, they are of a similar magnitude and qualitatively different from standard dual fermion theory. The high-frequency behaviour of the self-energy correction is - surprisingly - even dominated by corrections stemming from the three-particle vertex.

cond-mat.str-el↗

arXiv2017

Complementary views on electron spectra: From Fluctuation Diagnostics to real space correlations

We study the relation between the microscopic properties of a many-body system and the electron spectra, experimentally accessible by photoemission. In a recent paper [Phys. Rev. Lett. 114, 236402 (2015)], we introduced the "fluctuation diagnostics" approach, to extract the dominant wave vector dependent bosonic fluctuations from the electronic self-energy. Here, we first reformulate the theory in terms of fermionic modes, to render its connection with resonance valence bond (RVB) fluctuations more transparent. Secondly, by using a large-U expansion, where U is the Coulomb interaction, we relate the fluctuations to real space correlations. Therefore, it becomes possible to study how electron spectra are related to charge, spin, superconductivity and RVB-like real space correlations, broadening the analysis of an earlier work [Phys. Rev. B 89, 245130 (2014)]. This formalism is applied to the pseudogap physics of the two-dimensional Hubbard model, studied in the dynamical cluster approximation. We perform calculations for embedded clusters with up to 32 sites, having three inequivalent K-points at the Fermi surface. We find that as U is increased, correlation functions gradually attain values consistent with an RVB state. This first happens for correlation functions involving the antinodal point and gradually spreads to the nodal point along the Fermi surface. Simultaneously a pseudogap opens up along the Fermi surface. We relate this to a crossover from a Kondo-like state to an RVB-like localized cluster state and to the presence of RVB and spin fluctuations. These changes are caused by a strong momentum dependence in the cluster bath-couplings along the Fermi surface. We also show, from a more algorithmic perspective, how the time-consuming calculations in fluctuation diagnostics can be drastically simplified.

cond-mat.str-el↗

arXiv2017

Breakdown of traditional many-body theories for correlated electrons

Starting from the (Hubbard) model of an atom, we demonstrate that the uniqueness of the mapping from the interacting to the noninteracting Green's function, $G\to G_0$, is strongly violated, by providing numerous explicit examples of different $G_0$ leading to the same physical $G$. We argue that there are indeed infinitely many such $G_0$, with numerous crossings with the physical solution. We show that this rich functional structure is directly related to the divergence of certain classes of (irreducible vertex) diagrams, with important consequences for traditional many-body physics based on diagrammatic expansions. Physically, we ascribe the onset of these highly non-perturbative manifestations to the progressive suppression of the charge susceptibility induced by the formation of local magnetic moments and/or RVB states in strongly correlated electron systems.

cond-mat.str-el↗

arXiv2016

Non-perturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

We analyze the highly non-perturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory calculations at the two-particle level. By extending the results of Schäfer, et al. [Phys. Rev. Lett. 110, 246405 (2013)] we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. These divergence lines accumulate around the critical Mott endpoint in accordance with the interpretation as precursors of the MIT. By comparing our numerical data with analytical calculations of increasing complexity, such as for the disordered Binary Mixture and Falicov-Kimball (FK) models, as well as for the atomic limit (AL) case, (i) we identify two different kinds of divergences lines; (ii) we classify them in terms of the frequency-structure of the associated singular eigenvectors; (iii) we investigate their relation to the multiple branches in the Luttinger-Ward formalism. Moreover, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, unique energy scale $ν^*$ below which perturbation theory does no longer apply, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the non-perturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.

cond-mat.str-el↗

arXiv2016

Impact of nonlocal correlations over different energy scales: A Dynamical Vertex Approximation study

In this paper, we investigate how nonlocal correlations affect, selectively, the physics of correlated electrons over different energy scales, from the Fermi level to the band-edges. This goal is achieved by applying a diagrammatic extension of dynamical mean field theory (DMFT), the dynamical vertex approximation (D$Γ$A), to study several spectral and thermodynamic properties of the unfrustrated Hubbard model in two and three dimensions. Specifically, we focus first on the low-energy regime by computing the electronic scattering rate and the quasiparticle mass renormalization for decreasing temperatures at a fixed interaction strength. This way, we obtain a precise characterization of the several steps, through which the Fermi-liquid physics is progressively destroyed by nonlocal correlations. Our study is then extended to a broader energy range, by analyzing the temperature behavior of the kinetic and potential energy, as well as of the corresponding energy distribution functions. Our findings allow us to identify a smooth, but definite evolution of the nature of nonlocal correlations by increasing interaction: They either increase or decrease the kinetic energy w.r.t. DMFT depending on the interaction strength being weak or strong, respectively. This reflects the corresponding evolution of the ground state from a nesting-driven (Slater) to a superexchange-driven (Heisenberg) antiferromagnet (AF), whose fingerprints are, thus, recognizable in the spatial correlations of the paramagnetic phase. Finally, a critical analysis of our numerical results of the potential energy at the largest interaction allows us to identify possible procedures to improve the ladder-based algorithms adopted in the dynamical vertex approximation.

cond-mat.str-el↗

arXiv2016

Nonlocal correlations and spectral properties of the Falicov-Kimball model

We derive an analytical expression for the local two-particle vertex of the Falicov-Kimball model, including its dependence on all three frequencies, the full vertex and all reducible vertices. This allows us to calculate the self energy in diagrammatic extensions of dynamical mean field theory, specifically in the dual fermion and the one-particle irreducible approach. Non-local correlations are thence included and originate here from charge density wave fluctuations. At low temperatures and in two dimensions, they lead to a larger self energy contribution at low frequencies and a more insulating spectrum.

cond-mat.str-el↗

arXiv2016

Parquet decomposition calculations of the electronic self-energy

The parquet decomposition of the self-energy into classes of diagrams, those associated with specific scattering processes, can be exploited for different scopes. In this work, the parquet decomposition is used to unravel the underlying physics of non-perturbative numerical calculations. We show the specific example of dynamical mean field theory (DMFT) and its cluster extensions (DCA) applied to the Hubbard model at half-filling and with hole doping: These techniques allow for a simultaneous determination of two-particle vertex functions and self-energies, and hence, for an essentially "exact" parquet decomposition at the single-site or at the cluster level. Our calculations show that the self-energies in the underdoped regime are dominated by spin scattering processes, consistent with the conclusions obtained by means of the fluctuation diagnostics approach [Phys. Rev. Lett. 114, 236402 (2015)]. However, differently from the latter approach, the parquet procedure displays important changes with increasing interaction: Even for relatively moderate couplings, well before the Mott transition, singularities appear in different terms, with the notable exception of the predominant spin-channel. We explain precisely how these singularities, which partly limit the utility of the parquet decomposition, and - more generally - of parquet-based algorithms, are never found in the fluctuation diagnostics procedure. Finally, by a more refined analysis, we link the occurrence of the parquet singularities in our calculations to a progressive suppression of charge fluctuations and the formation of an RVB state, which are typical hallmarks of a pseudogap state in DCA.

cond-mat.str-el↗

arXiv2015

Dynamical vertex approximation in its parquet implementation: application to Hubbard nano-rings

We have implemented the dynamical vertex approximation (D$Γ$A) in its full parquet-based version to include spatial correlations on all length scales and in {\sl all} scattering channels. The algorithm is applied to study the electronic self-energies and the spectral properties of finite-size one-dimensional Hubbard models with periodic boundary conditions (nanoscopic Hubbard rings). From a methodological point of view, our calculations and their comparison to the results obtained within dynamical mean-field theory, plain parquet approximation, and the exact numerical solution, allow us to evaluate the performance of the D$Γ$A algorithm in the most challenging situation of low dimensions. From a physical perspective, our results unveil how non-local correlations affect the spectral properties of nanoscopic systems of various sizes in different regimes of interaction strength.

cond-mat.str-el↗

arXiv2014

Fate of the false Mott-Hubbard transition in two dimensions

We have studied the impact of non-local electronic correlations at all length scales on the Mott-Hubbard 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 is insulating at low enough temperatures for any (finite) interaction and no Mott-Hubbard transition is observed.

cond-mat.str-el↗

arXiv2014

Fluctuation diagnostics of the electron self-energy: Origin of the pseudogap physics

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.

cond-mat.str-el↗

arXiv2014

From infinite to two dimensions through the functional renormalization group

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

cond-mat.str-el↗

arXiv2013

One-particle irreducible functional approach - a new route to diagrammatic extensions of DMFT

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\GammaA) and the dual fermion (DF) scheme, yielding a consistent formulation of non-local 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\GammaA. To demonstrate the applicability and physical content of the 1PI approach, we compare the diagrammatics of 1PI, DF and D\GammaA, as well as the numerical results of these approaches for the half-filled Hubbard model in two dimensions.

cond-mat.str-el↗

arXiv2013

Divergent Precursors of the Mott-Hubbard Transition at the Two-Particle Level

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 non-perturbative 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. At low temperatures this occurs in the region where incoherent high-energy features emerge in the spectral function, while at high temperatures it is traceable up to the atomic-limit.

cond-mat.str-el↗

arXiv2011

Critical properties of the half-filled Hubbard model in three dimensions

By means of the dynamical vertex approximation (D$Γ$A) 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 ($T_N$) by $\sim 30%$ for the onset of antiferromagnetic order. Finally, we give a quantitative estimate of the deviation of the spectra between D$Γ$A and DMFT in different regions of the phase-diagram.

cond-mat.str-el↗