Dynamical Vertex Approximation Research Articles

Fermi and Luttinger Arcs: Two Concepts, Realized on One Surface.

We present an analytically solvable model for correlated electrons, which is able to capture the major Fermi surface modifications occurring in both hole- and electron-doped cuprates as a function of doping. The proposed Hamiltonian qualitatively reproduces the results of numerically demanding many-body calculations, here obtained using the dynamical vertex approximation. Our analytical theory provides a transparent description of a precise mechanism, capable of driving the formation of disconnected segments along the Fermi surface (the highly debated "Fermi arcs"), as well as the opening of a pseudogap in hole and electron doping. This occurs through a specific mechanism: The electronic states on the Fermi arcs remain intact, while the Fermi surface part where the gap opens transforms into a Luttinger arc.

Unconventional superconductivity without doping in infinite-layer nickelates under pressure

High-temperature unconventional superconductivity quite generically emerges from doping a strongly correlated parent compound, often (close to) an antiferromagnetic insulator. The recently developed dynamical vertex approximation is a state-of-the-art technique that has quantitatively predicted the superconducting dome of nickelates. Here, we apply it to study the effect of pressure in the infinite-layer nickelate SrxPr1−xNiO2. We reproduce the increase of the critical temperature (Tc) under pressure found in experiment up to 12 GPa. According to our results, Tc can be further increased with higher pressures. Even without Sr-doping the parent compound, PrNiO2, will become a high-temperature superconductor thanks to a strongly enhanced self-doping of the Ni dx2−y2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${d}_{{x}^{2}-{y}^{2}}$$\\end{document} orbital under pressure. With a maximal Tc of 100 K around 100 GPa, nickelate superconductors can reach that of the best cuprates.

Optimizing Superconductivity: From Cuprates via Nickelates to Palladates.

Motivated by cuprate and nickelate superconductors, we perform a comprehensive study of the superconducting instability in the single-band Hubbard model. We calculate the spectrum and superconducting transition temperature T_{c} as a function of filling and Coulomb interaction for a range of hopping parameters, using the dynamical vertex approximation. We find the sweet spot for high T_{c} to be at intermediate coupling, moderate Fermi surface warping, and low hole doping. Combining these results with first principles calculations, neither nickelates nor cuprates are close to this optimum within the single-band description. Instead, we identify some palladates, notably RbSr_{2}PdO_{3} and A_{2}^{'}PdO_{2}Cl_{2} (A^{'}=Ba_{0.5}La_{0.5}), to be virtually optimal, while others, such as NdPdO_{2}, are too weakly correlated.

Ab initio materials design of superconductivity in d9 nickelates

Motivated by the recent theoretical materials design of superconducting d9 nickelates for which the charge transfer from the NiO2 layer to the block layer is completely suppressed [M. Hirayama et al., Phys. Rev. B 101, 075107 (2020)], we perform a calculation based on the dynamical vertex approximation and obtain the phase diagram of RbCa2NiO3 and A2NiO2Br2, where A is a cation with a valence of 2.5+. We show that the phase diagram of these nickelates exhibits the same essential features as those found in cuprates. Namely, superconductivity appears upon hole-doping into an antiferromagnetic Mott insulator, and the superconducting transition temperature shows a dome-like shape. This demonstrates that the electron correlations play an essential role in nickelate superconductors, and we can control them by changing block layers.

Consistency of potential energy in the dynamical vertex approximation

In the last decades, dynamical mean-field theory (DMFT) and its diagrammatic extensions have been successfully applied to describe local and nonlocal correlation effects in correlated electron systems. Unfortunately, except for the exact solution, it is impossible to fulfill both the Pauli principle and conservation laws at the same time. Consequently, fundamental observables such as the kinetic and potential energies are ambiguously defined. In this work, we propose an approach to overcome the ambiguity in the calculation of the potential energy within the ladder dynamical vertex approximation (D$\Gamma$A) by introducing an effective mass renormalization parameter in both the charge and the spin susceptibility of the system. We then apply our method to the half-filled single-band Hubbard model on a three-dimensional bipartite cubic lattice. We find that: (i) at weak-to-intermediate coupling, a reasonable modification of the transition temperature $\TN$ to the antiferromagnetically ordered state with respect to previous ladder D$\Gamma$A calculations without charge renormalization. This is in good agreement with dual fermion and Monte Carlo results; (ii) the renormalization of charge fluctuations in our new approach leads to a unique value for the potential energy which is substantially lower than corresponding ones from DMFT and non-self-consistent ladder D$\Gamma$A; and (iii) the hierarchy of the kinetic energies between the DMFT and the ladder D$\Gamma$A in the weak coupling regime is restored by the consideration of charge renormalization.

Particle-hole asymmetric lifetimes promoted by nonlocal spin and orbital fluctuations in SrVO3 monolayers

The two-dimensional nature of engineered transition-metal ultrathin oxide films offers a large playground of yet to be fully understood physics. Here, we study pristine ${\mathrm{SrVO}}_{3}$ monolayers that have recently been predicted to display a variety of magnetic and orbital orders. We find that nonlocal magnetic (orbital) fluctuations lead to a strong (weak to moderate) momentum differentiation in the self-energy, particularly in the scattering rate. In the one-band 2D Hubbard model, momentum selectivity on the Fermi surface (``$k={k}_{F}$'') is known to lead to pseudogap physics. Here instead, in the multiorbital case, we evidence a differentiation between momenta on the occupied (``$k<{k}_{F}$'') and the unoccupied side (``$k>{k}_{F}$'') of the Fermi surface. Based on the dynamical vertex approximation, and introducing a ``binaural fluctuation diagnostics'' tool, we advance the understanding of spectral signatures of nonlocal fluctuations. Our work calls to (re)examine ultrathin oxide films and interfaces with methods beyond dynamical mean-field theory and may point to correlation-enhanced thermoelectric effects.

Magnetic correlations in infinite-layer nickelates: An experimental and theoretical multimethod study

We report a comprehensive study of magnetic correlations in ${\mathrm{LaNiO}}_{2}$, a parent compound of the recently discovered family of infinite-layer (IL) nickelate superconductors, using multiple experimental and theoretical methods. Our specific heat, muon-spin rotation ($\ensuremath{\mu}\mathrm{SR}$), and magnetic susceptibility measurements on polycrystalline ${\mathrm{LaNiO}}_{2}$ show that long-range magnetic order remains absent down to 2 K. Nevertheless, we detect residual entropy in the low-temperature specific heat, which is compatible with a model fit that includes paramagnon excitations. The $\ensuremath{\mu}\mathrm{SR}$ and low-field static and dynamic magnetic susceptibility measurements indicate the presence of short-range magnetic correlations and glassy spin dynamics, which we attribute to local oxygen nonstoichiometry in the average infinite-layer crystal structure. This glassy behavior can be suppressed in strong external fields, allowing us to extract the intrinsic paramagnetic susceptibility. Remarkably, we find that the intrinsic susceptibility shows non-Curie-Weiss behavior at high temperatures, in analogy to doped cuprates that possess robust nonlocal spin fluctuations. The distinct temperature dependence of the intrinsic susceptibility of ${\mathrm{LaNiO}}_{2}$ can be theoretically understood by a multimethod study of the single-band Hubbard model in which we apply complementary cutting-edge quantum many-body techniques (dynamical mean-field theory, cellular dynamical mean-field theory, and the dynamical vertex approximation) to investigate the influence of both short- and long-ranged correlations. Our results suggest a profound analogy between the magnetic correlations in parent (undoped) IL nickelates and doped cuprates.

Magnetic Properties and Pseudogap Formation in Infinite-Layer Nickelates: Insights From the Single-Band Hubbard Model

We study the magnetic and spectral properties of a single-band Hubbard model for the infinite-layer nickelate compound LaNiO2. As spatial correlations turn out to be the key ingredient for understanding its physics, we use two complementary extensions of the dynamical mean-field theory to take them into account: the cellular dynamical mean-field theory and the dynamical vertex approximation. Additionally to the systematic analysis of the doping dependence of the non-Curie-Weiss behavior of the uniform magnetic susceptibility, we provide insight into its relation to the formation of a pseudogap regime by the calculation of the one-particle spectral function and the magnetic correlation length. The latter is of the order of a few lattice spacings when the pseudogap opens, indicating a strong-coupling pseudogap formation in analogy to cuprates.

Phase Diagram of Nickelate Superconductors Calculated by Dynamical Vertex Approximation

We review the electronic structure of nickelate superconductors with and without effects of electronic correlations. As a minimal model, we identify the one-band Hubbard model for the Ni 3dx2−y2 orbital plus a pocket around the A-momentum. The latter, however, merely acts as a decoupled electron reservoir. This reservoir makes a careful translation from nominal Sr-doping to the doping of the one-band Hubbard model mandatory. Our dynamical mean-field theory calculations, in part already supported by the experiment, indicate that the Γ pocket, Nd 4f orbitals, oxygen 2p, and the other Ni 3d orbitals are not relevant in the superconducting doping regime. The physics is completely different if topotactic hydrogen is present or the oxygen reduction is incomplete. Then, a two-band physics hosted by the Ni 3dx2−y2 and 3d3z2−r2orbitals emerges. Based on our minimal modeling, we calculated the superconducting Tc vs. Sr-doping x phase diagram prior to the experiment using the dynamical vertex approximation. For such a notoriously difficult to determine quantity as Tc, the agreement with the experiment is astonishingly good. The prediction that Tc is enhanced with pressure or compressive strain has been confirmed experimentally as well. This supports that the one-band Hubbard model plus an electron reservoir is the appropriate minimal model.

Dynamical vertex approximation for many-electron systems with spontaneously broken SU(2) symmetry

We generalize the formalism of the dynamical vertex approximation (D$\Gamma$A) -- a diagrammatic extension of the dynamical mean-field theory (DMFT)-- to treat magnetically ordered phases. To this aim, we start by concisely illustrating the many-electron formalism for performing ladder resummations of Feynman diagrams in systems with broken SU(2)-symmetry, associated to ferromagnetic (FM) or antiferromagnetic (AF) order. We then analyze the algorithmic simplifications introduced by taking the local approximation of the two-particle irreducible vertex functions in the Bethe-Salpeter equations, which defines the ladder implementation of D$\Gamma$A for magnetic systems. The relation of this assumption with the DMFT limit of large coordination-number/ high-dimensions is explicitly discussed. As a last step, we derive the expression for the ladder D$\Gamma$A self-energy in the FM- and AF-ordered phases of the Hubbard model. The physics emerging in the AF-ordered case is explicitly illustrated by means of approximated calculations based on a static mean-field input for the D$\Gamma$A equations. The results obtained capture fundamental aspects of both metallic and insulating ground states of two-dimensional antiferromagnets, providing a reliable compass for future, more extensive applications of our approach. Possible routes to further develop diagrammatic-based treatments of magnetic phases in correlated electron systems are briefly outlined in the conclusions.

Anisotropy of electronic correlations: On the applicability of local theories to layered materials

Besides the chemical constituents, it is the lattice geometry that controls the most important material properties. In many interesting compounds, the arrangement of elements leads to pronounced anisotropies, which reflect into a varying degree of quasi two-dimensionality of their low-energy excitations. Here, we start by classifying important families of correlated materials according to a simple measure for the tetragonal anisotropy of their ab initio electronic (band) structure. Second, we investigate the impact of a progressively large anisotropy in driving the non-locality of many-body effects. To this end, we tune the Hubbard model from isotropic cubic in three dimensions to the two-dimensional limit and analyze it using the dynamical vertex approximation. For sufficiently isotropic hoppings, we find the self-energy to be well separable into a static non-local and a dynamical local contribution. While the latter could potentially be obtained from dynamical mean-field approaches, we find the former to be non-negligible in all cases. Further, by increasing the model-anisotropy, we quantify the degree of quasi two-dimensionality which causes this "space-time separation" to break down. Our systematic analysis improves the general understanding of electronic correlations in anisotropic materials, heterostructures and ultra-thin films, and provides useful guidance for future realistic studies.

Self-consistent ladder dynamical vertex approximation

We present and implement a self-consistent D$\Gamma$A approach for multi-orbital models and ab initio materials calculations. It is applied to the one-band Hubbard model at various interaction strengths with and without doping, to the two-band Hubbard model with two largely different bandwidths, and to SrVO$_3$. The self-energy feedback reduces critical temperatures compared to dynamical mean-field theory, even to zero temperature in two-dimensions. Compared to a one-shot, non-self-consistent calculation the non-local correlations are significantly reduced when they are strong. In case non-local correlations are weak to moderate as for SrVO$_3$, one-shot calculations are sufficient.

Statistical error estimates in dynamical mean-field theory and extensions thereof

We employ the jackknife algorithm to analyze the propagation of the statistical quantum Monte Carlo error through the Bethe--Salpeter equation. This allows us to estimate the error of dynamical mean-field theory calculations of the susceptibility and of dynamical vertex approximation calculations of the self-energy. We find that the different frequency components of the susceptibility are uncorrelated, whereas those of the self-energy are correlated. For improving the quality of the correlation matrix taking sufficiently many jackknife bins is key, while for reducing the standard error of the mean sufficiently many Monte Carlo measurements are necessary. We furthermore show that even in the case of the self-energy, the finite covariance does not have a sizable influence on the analytic continuation.

The AbinitioD[formula omitted]A Project v1.0: Non-local correlations beyond and susceptibilities within dynamical mean-field theory

The ab initio extension of the dynamical vertex approximation (DΓA) method allows for realistic materials calculations that include non-local correlations beyond GW and dynamical mean-field theory. Here, we discuss the AbinitioDΓA algorithm, its implementation and usage in detail, and make the program package available to the scientific community. Program summaryProgram Title: AbinitioDΓAProgram Files doi:http://dx.doi.org/10.17632/h3k3fg6szb.1Licensing provisions: GNU General Public License v3Programming language:Fortran95 and PythonRequired dependencies:MPI, LAPACK, BLAS, HDF5(≥1.8.12), Python(≥2.7), h5py(≥2.5.0), numpy(≥1.9.1)Optional dependencies:pip, matplotlib(≥1.5.1), scipy(≥0.14.0)Supplementary material: Test case files and step-by-step instructions.Nature of problem: Realistic materials calculations including non-local correlations beyond dynamical mean-field theory (DMFT) as well as non-local interactions. Solving the Bethe–Salpeter equation for multiple orbitals. Determining momentum-resolved susceptibilities in DMFT.Solution method:Ab initio dynamical vertex approximation: starting from the local two-particle vertex and constructing from it the local DMFT correlations, the GW diagrams, and further non-local correlations, e.g., spin fluctuations. Efficient solution of the Bethe–Salpeter equation, avoiding divergencies in the irreducible vertex in the particle-hole channel by reformulating the problem in terms of the full vertex. Parallelization with respect to the bosonic frequency and transferred momentum.Additional comments including restrictions and unusual features: As input, a Hamiltonian derived, e.g., from density functional theory and a DMFT solution thereof is needed including a local two-particle vertex calculated at DMFT self-consistency. Hitherto the AbinitioDΓA program package is restricted to SU(2) symmetric problems. A so-called λ correction or self-consistency is not yet implemented in the AbinitioDΓA code. Susceptibilities are so far only calculated within DMFT, not the dynamical vertex approximation.

Quantum Criticality in the Two-Dimensional Periodic Anderson Model.

We study the phase diagram and quantum critical region of one of the fundamental models for electronic correlations: the periodic Anderson model. Employing the recently developed dynamical vertex approximation, we find a phase transition between a zero-temperature antiferromagnetic insulator and a Kondo insulator. In the quantum critical region, we determine a critical exponent γ=2 for the antiferromagnetic susceptibility. At higher temperatures, we have free spins with γ=1 instead, whereas at lower temperatures, there is an even stronger increase and suppression of the susceptibility below and above the quantum critical point, respectively.

Efficient evaluation of the polarization function in dynamical mean-field theory

The dynamical susceptibility of strongly correlated electronic systems can be calculated within the framework of the dynamical mean-field theory (DMFT). The required measurement of the four-point vertex of the auxiliary impurity model is however costly and restricted to a finite grid of Matsubara frequencies, leading to a cutoff error. It is shown that the propagation of this error to the lattice response function can be minimized by virtue of an exact decomposition of the DMFT polarization function into local and nonlocal parts. The former is measured directly by the impurity solver, while the latter is given in terms of a ladder equation for the Hedin vertex that features an unprecedentedly fast decay of frequency summations compared to previous calculation schemes, such as the one of the dual boson approach. At strong coupling the local approximation of the TRILEX approach is viable, but vertex corrections to the polarization should be dropped on equal footing to recover the correct prefactor of the effective exchange. In finite dimensions the DMFT susceptibility exhibits spurious mean-field criticality, therefore, a two-particle self-consistent and frequency-dependent correction term is introduced, similar to the Moriya-$\lambda$ correction of the dynamical vertex approximation. Applications to the two- and three-dimensional Hubbard models on the square and cubic lattices show that the expected critical behavior near an antiferromagnetic instability is recovered.

Dynamical vertex approximation for the attractive Hubbard model

In this work, we adapt the formalism of the dynamical vertex approximation (D$\Gamma$A), a diagrammatic approach including many-body correlations beyond the dynamical mean-field theory, to the case of attractive onsite interactions. We start by exploiting the ladder approximation of the D$\Gamma$A scheme, in order to derive the corresponding equations for the non-local self-energy and vertex functions of the attractive Hubbard model. Second, we prove the validity of our derivation by showing that the results obtained in the particle-hole symmetric case fully preserve the exact mapping between the attractive and the repulsive models. It will be shown, how this property can be related to the structure of the ladders, which makes our derivation applicable for any approximation scheme based on ladder diagrams. Finally, we apply our D$\Gamma$A algorithm to the attractive Hubbard model in three dimensions, for different fillings and interaction values. Specifically, we focus on the parameters region in the proximity of the second-order transition to the superconducting and charge-density wave phases, respectively, and calculate (i) their phase-diagrams, (ii) their critical behavior, as well as (iii) the effects of the strong non-local correlations on the single-particle properties.

Why the critical temperature of high- Tc cuprate superconductors is so low: The importance of the dynamical vertex structure

To fathom the mechanism of high-temperature ($T_{\rm c}$) superconductivity, the dynamical vertex approximation (D$\Gamma$A) is evoked for the two-dimensional repulsive Hubbard model. After showing that our results well reproduce the cuprate phase diagram with a reasonable $T_{\rm c}$ and dome structure, we keep track of the scattering processes that primarily affect $T_{\rm c}$. We find that local particle-particle diagrams significantly screen the bare interaction at low frequencies, which in turn suppresses antiferromagnetic spin fluctuations and hence the pairing interaction. Thus we identify dynamical vertex corrections as one of the main oppressors of $T_{\rm c}$, which may provide a hint toward higher $T_{\rm c}$'s.

Precursors of the insulating state in the square-lattice Hubbard model

We study the two-dimensional square lattice Hubbard model for small to moderate interaction strengths $1\leq U/t\leq 4$ by means of the ladder dual fermion approach. The non-local correlations beyond dynamical mean-field theory lower the potential energy, lead to a maximum in the uniform susceptibility and induce a pseudogap in the density of states. While the self-energy exhibits precursors of a possible insulating phase linked to the appearance of long-range fluctuations, the metallic phase persists within the accessible temperature range. Finite-size effects affect results qualitatively. Upper bounds on the crossover temperature are found to be significantly lower than previously reported dynamical vertex approximation results at $U/t=1$.

Towards ab initio Calculations with the Dynamical Vertex Approximation

While key effects of the many-body problem---such as Kondo and Mott physics---can be understood in terms of on-site correlations, non-local fluctuations of charge, spin, and pairing amplitudes are at the heart of the most fascinating and unresolved phenomena in condensed matter physics. Here, we review recent progress in diagrammatic extensions to dynamical mean-field theory for ab initio materials calculations. We first recapitulate the quantum field theoretical background behind the two-particle vertex. Next we discuss latest algorithmic advances in quantum Monte Carlo simulations for calculating such two-particle quantities using worm sampling and vertex asymptotics, before giving an introduction to the ab initio dynamical vertex approximation (AbinitioD$\Gamma$A). Finally, we highlight the potential of AbinitioD$\Gamma$A by detailing results for the prototypical correlated metal SrVO$_3$.