Nonexistence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for hubbard-like models

A Similarity Renormalization Group Approach to Green's Function Methods

The family of Green's function methods based on the GW approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study [Monino and Loos J. Chem. Phys. 2022, 156, 231101.] has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first-principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent GW (qsGW) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qsGW calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.

Nonlocal Kondo effect and two-fluid picture revealed in an exactly solvable model

Understanding the nature of local-itinerant transition of strongly correlated electrons is one of the central problems in condensed matter physics. Heavy fermion systems describe the f-electron delocalization through Kondo interactions with conduction electrons. Tremendous efforts have been devoted to the so-called Kondo-destruction scenario, which predicts a dramatic local-to-itinerant quantum phase transition of f-electrons at zero temperature. On the other hand, two-fluid behaviors have been observed in many materials, suggesting coexistence of local and itinerant f-electrons over a broad temperature range but lacking a microscopic theoretical description. To elucidate this fundamental issue, here we propose an exactly solvable Kondo-Heisenberg model in which the spins are defined in the momentum space and the k-space Kondo interaction corresponds to a highly nonlocal spin scattering in the coordinate space. Its solution reveals a continuous evolution of the Fermi surfaces with Kondo interaction and two-fluid behaviors similar to those observed in real materials. The electron density violates the usual Luttinger's theorem, but follows a generalized one allowing for partially enlarged Fermi surfaces due to partial Kondo screening in the momentum space. Our results highlight the consequence of nonlocal Kondo interaction relevant for strong quantum fluctuation regions and provide important insight into the microscopic description of two-fluid phenomenology in heavy fermion systems.

Vertex-Based Diagrammatic Treatment of Light-Matter-Coupled Systems

We propose a diagrammatic Monte Carlo approach for quantum impurity models, which can be regarded as a generalization of the strong-coupling expansion for fermionic impurity models. The algorithm is based on a self-consistently computed three-point vertex and a stochastically sampled four-point vertex, and it allows one to obtain numerically exact results in a wide parameter regime. The performance of the algorithm is demonstrated with applications to a spin-boson model representing an emitter in a waveguide. As a function of the coupling strength, the spin exhibits a delocalization-localization crossover at low temperatures, signaling a qualitative change in the real-time relaxation. In certain parameter regimes, the response functions of the emitter coupled to the electromagnetic continuum can be described by an effective Rabi model with appropriately defined parameters. We also discuss the spatial distribution of the photon density around the emitter.

Homotopic Action: A Pathway to Convergent Diagrammatic Theories

The major obstacle preventing Feynman diagrammatic expansions from accurately solving many-fermion systems in strongly correlated regimes is the series slow convergence or divergence problem. Several techniques have been proposed to address this issue: series resummation by conformal mapping, changing the nature of the starting point of the expansion by shifted action tools, and applying the homotopy analysis method to the Dyson-Schwinger equation. They emerge as dissimilar mathematical procedures aimed at different aspects of the problem. The proposed homotopic action offers a universal and systematic framework for unifying the existing-and generating new-methods and ideas to formulate a physical system in terms of a convergent diagrammatic series. It eliminates the need for resummation, allows one to introduce effective interactions, enables a controlled ultraviolet regularization of continuous-space theories, and reduces the intrinsic polynomial complexity of the diagrammatic Monte Carlo method. We illustrate this approach by an application to the Hubbard model.

How to read between the lines of electronic spectra: the diagnostics of fluctuations in strongly correlated electron systems

While calculations and measurements of single-particle spectral properties often offer the most direct route to study correlated electron systems, the underlying physics may remain quite elusive, if information at higher particle levels is not explicitly included. Here, we present a comprehensive overview of the different approaches which have been recently developed and applied to identify the dominant two-particle scattering processes controlling the shape of the one-particle spectral functions and, in some cases, of the physical response of the system. In particular, we will discuss the underlying general idea, the common threads and the specific peculiarities of all the proposed approaches. While all of them rely on a selective analysis of the Schwinger-Dyson (or the Bethe-Salpeter) equation, the methodological differences originate from the specific two-particle vertex functions to be computed and decomposed. Finally, we illustrate the potential strength of these methodologies by means of their applications the two-dimensional Hubbard model, and we provide an outlook over the future perspective and developments of this route for understanding the physics of correlated electrons.

Fingerprints of the Local Moment Formation and its Kondo Screening in the Generalized Susceptibilities of Many-Electron Problems

We identify the precise hallmarks of the local magnetic moment formation and its Kondo screening in the frequency structure of the generalized charge susceptibility. The sharpness of our identification even pinpoints an alternative criterion to determine the Kondo temperature of strongly correlated systems on the two-particle level, which only requires calculations at the lowest Matsubara frequency. We showcase its strength by applying it to the single impurity and the periodic Anderson model as well as to the Hubbard model. Our results represent a significant progress for the general understanding of quantum field theory at the two-particle level and allow for tracing the limits of the physics captured by perturbative approaches for correlated systems.

Attractive Effect of a Strong Electronic Repulsion: The Physics of Vertex Divergences

While the breakdown of the perturbation expansion for the many-electron problem has several formal consequences, here we unveil its physical effect: flipping the sign of the effective electronic interaction in specific scattering channels. By decomposing local and uniform susceptibilities of the Hubbard model via their spectral representations, we prove how entering the nonperturbative regime causes an enhancement of the charge response, ultimately responsible for the phase-separation instabilities close to the Mott metal-insulator transition. Our analysis opens a new route for understanding phase transitions in the nonperturbative regime and clarifies why attractive effects emerging from a strong repulsion can induce phase separations but not s-wave pairing or charge-density wave instabilities.

Bethe-Salpeter Equation at the Critical End Point of the Mott Transition

Strong repulsive interactions between electrons can lead to a Mott metal-insulator transition. The dynamical mean-field theory (DMFT) explains the critical end point and the hysteresis region usually in terms of single-particle concepts, such as the spectral function and the quasiparticle weight. In this Letter, we reconsider the critical end point of the metal-insulator transition on the DMFT's two-particle level. We show that the relevant eigenvalue and eigenvector of the nonlocal Bethe-Salpeter kernel in the charge channel provide a unified picture of the hysteresis region and of the critical end point of the Mott transition. In particular, they simultaneously explain the thermodynamics of the hysteresis region and the iterative stability of the DMFT equations. This analysis paves the way for a deeper understanding of phase transitions in correlated materials.

Wave Function Perspective and Efficient Truncation of Renormalized Second-Order Perturbation Theory

We present an approach to a renormalized second-order Green's function perturbation theory (GF2), which avoids all dependency on continuous variables, grids, or explicit Green's functions and is instead formulated entirely in terms of static quantities and wave functions. Correlation effects from MP2 diagrams are iteratively incorporated to modify the underlying spectrum of excitations by coupling the physical system to fictitious auxiliary degrees of freedom, allowing for single-particle orbitals to delocalize into this additional space. The overall approach is shown to be rigorously O[N⁵], after an appropriate compression of this auxiliary space. This is achieved via a novel scheme, which ensures that a desired number of moments of the underlying occupied and virtual spectra are conserved in the compression, allowing a rapid and systematically improvable convergence to the limit of the effective dynamical resolution. The approach is found to then allow for the qualitative description of stronger correlation effects, avoiding the divergences of MP2, as well as its orbital-optimized version. On application to the G1 test set, we find that modification up to only the third spectral moment of the underlying spectrum from which the double excitations are built are required for accurate energetics, even in strongly correlated regimes. This is beyond simple self-consistent changes to the density matrix of the system but far from requiring a description of the full dynamics of the frequency-dependent self-energy.

A combined variational and diagrammatic quantum Monte Carlo approach to the many-electron problem

Two of the most influential ideas developed by Richard Feynman are the Feynman diagram technique and his variational approach. Here we show that combining both, and introducing a diagrammatic quantum Monte Carlo method, results in a powerful and accurate solver to the generic solid state problem, in which a macroscopic number of electrons interact by the long range Coulomb repulsion. We apply it to the quintessential problem of solid state, the uniform electron gas, which is at the heart of the density functional theory success in describing real materials, yet it has not been adequately solved for over 90 years. Our method allows us to calculate numerically exact momentum and frequency resolved spin and charge response functions. This method can be applied to a number of moderately interacting electron systems, including models of realistic metallic and semiconducting solids.

Developments of new computational methods for interacting electron systems have important implications for predictions of material properties. Chen and Haule combine variational techniques with diagrammatic quantum Monte Carlo to perform numerically exact calculations of electron gas properties, including the spin susceptibility and the dielectric function.

The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy

The GW approximation in electronic structure theory has become a widespread tool for predicting electronic excitations in chemical compounds and materials. In the realm of theoretical spectroscopy, the GW method provides access to charged excitations as measured in direct or inverse photoemission spectroscopy. The number of GW calculations in the past two decades has exploded with increased computing power and modern codes. The success of GW can be attributed to many factors: favorable scaling with respect to system size, a formal interpretation for charged excitation energies, the importance of dynamical screening in real systems, and its practical combination with other theories. In this review, we provide an overview of these formal and practical considerations. We expand, in detail, on the choices presented to the scientist performing GW calculations for the first time. We also give an introduction to the many-body theory behind GW, a review of modern applications like molecules and surfaces, and a perspective on methods which go beyond conventional GW calculations. This review addresses chemists, physicists and material scientists with an interest in theoretical spectroscopy. It is intended for newcomers to GW calculations but can also serve as an alternative perspective for experts and an up-to-date source of computational techniques.

Band-Edge Quasiparticles from Electron-Phonon Coupling and Resistivity Saturation

We address the problem of resistivity saturation observed in materials such as the A-15 compounds. To do so, we calculate the resistivity for the Hubbard-Holstein model in infinite spatial dimensions to second order in on-site repulsion U≤D and to first order in (dimensionless) electron-phonon coupling strength λ≤0.5, where D is the half bandwidth. We identify a unique mechanism to obtain two parallel quantum conducting channels: low-energy and band-edge high-energy quasi-particles. We identify the source of the hitherto unremarked high-energy quasiparticles as a positive slope in the frequency dependence of the real part of the electron self-energy. In the presence of phonons, the self-energy grows linearly with the temperature at high T, causing the resistivity to saturate. As U is increased, the saturation temperature is pushed to higher values, offering a mechanism by which electron correlations destroy saturation.

Topological order in the pseudogap metal

We compute the electronic Green's function of the topologically ordered Higgs phase of a SU(2) gauge theory of fluctuating antiferromagnetism on the square lattice. The results are compared with cluster extensions of dynamical mean field theory, and quantum Monte Carlo calculations, on the pseudogap phase of the strongly interacting hole-doped Hubbard model. Good agreement is found in the momentum, frequency, hopping, and doping dependencies of the spectral function and electronic self-energy. We show that lines of (approximate) zeros of the zero-frequency electronic Green's function are signs of the underlying topological order of the gauge theory and describe how these lines of zeros appear in our theory of the Hubbard model. We also derive a modified, nonperturbative version of the Luttinger theorem that holds in the Higgs phase.

Many-body perturbation theory and non-perturbative approaches: screened interaction as the key ingredient

Many-body perturbation theory is often formulated in terms of an expansion in the dressed instead of the bare Green's function, and in the screened instead of the bare Coulomb interaction. However, screening can be calculated on different levels of approximation, and it is important to define what is the most appropriate choice. We explore this question by studying a zero-dimensional model (so called 'one-point model') that retains the structure of the full equations. We study both linear and non-linear response approximations to the screening. We find that an expansion in terms of the screening in the random phase approximation is the most promising way for an application in real systems. Moreover, by making use of the nonperturbative features of the Kadanoff-Baym equation for the one-body Green's function, we obtain an approximate solution in our model that is very promising, although its applicability to real systems has still to be explored.

Variational structure of Luttinger-Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory

The Luttinger-Ward functional was proposed more than five decades ago and has been used to formally justify most practically used Green's function methods for quantum many-body systems. Nonetheless, the very existence of the Luttinger-Ward functional has been challenged by recent theoretical and numerical evidence. We provide a rigorously justified Luttinger-Ward formalism, in the context of Euclidean lattice field theory. Using the Luttinger-Ward functional, the free energy can be variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.

Nonperturbative Series Expansion of Green's Functions: The Anatomy of Resonant Inelastic X-Ray Scattering in the Doped Hubbard Model

We present a nonperturbative, divergence-free series expansion of Green's functions using effective operators. The method is especially suited for computing correlators of complex operators as a series of correlation functions of simpler forms. We apply the method to study low-energy excitations in resonant inelastic x-ray scattering (RIXS) in doped one- and two-dimensional single-band Hubbard models. The RIXS operator is expanded into polynomials of spin, density, and current operators weighted by fundamental x-ray spectral functions. These operators couple to different polarization channels resulting in simple selection rules. The incident photon energy dependent coefficients help to pinpoint main RIXS contributions from different degrees of freedom. We show in particular that, with parameters pertaining to cuprate superconductors, local spin excitation dominates the RIXS spectral weight over a wide doping range in the cross-polarization channel.

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 function, G→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 nonperturbative manifestations to the progressive suppression of the charge susceptibility induced by the formation of local magnetic moments and/or resonating valence bond (RVB) states in strongly correlated electron systems.