One-body Green Function Research Articles

Total Energy beyond GW: Exact Results and Guidelines for Approximations.

The total energy and electron addition and removal spectra can, in principle, be obtained exactly from the one-body Green's function (GF). In practice, the GF is obtained from an approximate self-energy. In the framework of many-body perturbation theory, we derive different expressions that are based on an approximate self-energy, but that yield nevertheless, in principle, the exact exchange-correlation contribution to the total energy for any interaction strength. Response functions play a crucial role, which explains why, for example, ingredients from time-dependent density functional theory can be used to build these approximate self-energies. We show that the key requirement for obtaining exact results is the consistent combination of ingredients. Also when further approximations are made, as it is necessary in practice, this consistency remains the key to obtain good results. All findings are illustrated using the exactly solvable symmetric Hubbard dimer.

The dilute interacting Bose gas: comparisons between Feynman diagrammatic expansion and the hierarchy equations for the imaginary time Green functions

We analyze the correspondence between two formalisms describing an interacting Bose gas, namely the standard Feynman diagrammatic expansion on the one hand, and the hierarchy equations for the imaginary-time Green functions on the other hand. We show that the Hartree–Fock approximation, as well as its first corrections at low density derived by Baym et al (2001 Eur. Phys. J. B 24 107–24), can be equivalently formulated in both formalisms. Within the hierarchy approach, the two-body correlations are expressed at lowest order in the interactions in terms of the full one-body Green function. This sheds light on the physical content of the corrections to the Hartree–Fock theory. Moreover, this representation of can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.

Determination of the Fermi surface by charge density correlations

The Fermi surface topology plays a crucial role in the study of high-temperature superconductivity cuprates. The conventional method for determining the Fermi surface is the maximum spectral intensity method, which involves numerical analytical continuation of Matsubara data for the one-body Green's function. However, the numerical analytical continuation is sensitive to the noise or the precision of the Matsubara data, and hence is not always reliable. With this in mind, we propose a simple and specific notion as a reference for the Fermi surface. It is the derivative of the momentum distribution function ${n}_{\mathbit{k}}$ with respect to the chemical potential $\ensuremath{\mu}$. Our analysis of the noninteracting system shows that the momentum, at which $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ takes the maximum value, constitutes a surface, which coincides with the Fermi surface. The relationship between $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ and the Fermi surface in general cases is also analyzed. In order to numerically demonstrate this relationship, we calculate and compare on the two-dimensional Hubbard model within the HGW method. The results show that, at least in the weak and intermediate coupling regimes, using the notion of $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ to determine the Fermi surface is reliable. We further studied the cases with model parameters standing for realistic materials, and found the surface determined by $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ exhibits topological transition with the charge filling, similar to the Fermi surface in the cuprates. We believe that using the derivative of the momentum distribution function with respect to the chemical potential to determine the Fermi surface is a reasonable, efficient, and potentially valuable approach.

Photoemission spectral functions from the three-body Green's function

We present an original strategy for the calculation of direct and inverse photo-emission spectra from first principles. The main goal is to go beyond the standard Green's function approaches, such as the $GW$ method, in order to find a good description not only of the quasiparticles but also of the satellite structures, which are of particular importance in strongly correlated materials. To this end we use as a key quantity the three-body Green's function, or, more precisely, its hole-hole-electron and electron-electron-hole parts, and we show how the one-body Green's function, and hence the corresponding spectral function, can be retrieved from it. We show that, contrary to the one-body Green's function, information about satellites is already present in the non-interacting three-body Green's function. Therefore, simple approximations to the three-body self-energy, which is defined by the Dyson equation for the three-body Green's function and which contains many-body effects, can still yield accurate spectral functions. In particular, the self-energy can be chosen to be static which could simplify a self-consistent solution of the Dyson equation. We give a proof of principle of our strategy by applying it to the Hubbard dimer, for which the exact self-energy is available.

Diagonal Padé approximant of the one-body Green's function: A study on Hubbard rings

Pad\'e approximants to the many-body Green's function can be built by rearranging terms of its perturbative expansion. The hypothesis that the best use of a finite number of terms of such an expansion is given by the subclass of diagonal Pad\'e approximants is here tested, and largely confirmed, on a solvable model system, namely the Hubbard ring for a variety of site numbers, fillings and interaction strengths.

Spectral properties from Matsubara Green's function approach: Application to molecules

We present results for many-body perturbation theory for the one-body Green's function at finite temperatures using the Matsubara formalism. Our method relies on the accurate representation of the single-particle states in standard Gaussian basis sets, allowing to efficiently compute, among other observables, quasiparticle energies and Dyson orbitals of atoms and molecules. In particular, we challenge the second-order treatment of the Coulomb interaction by benchmarking its accuracy for a well-established test set of small molecules, which includes also systems where the usual Hartree-Fock treatment encounters difficulties. We discuss different schemes how to extract quasiparticle properties and assess their range of applicability. With an accurate solution and compact representation, our method is an ideal starting point to study electron dynamics in time-resolved experiments by the propagation of the Kadanoff-Baym equations.

Many-body computations by stochastic sampling in Hartree-Fock-Bogoliubov space

We describe the computational ingredients for an approach to treat interacting fermion systems in the presence of pairing fields, based on path-integrals in the space of Hartree-Fock-Bogoliubov (HFB) wave functions. The path-integrals can be evaluated by Monte Carlo, via random walks of HFB wave functions whose orbitals evolve stochastically. The approach combines the advantage of HFB theory in paired fermion systems and many-body quantum Monte Carlo (QMC) techniques. The properties of HFB states, written in the form of either product states or Thouless states, are discussed. The states are shown to be propagated by generalized one-body operators. The states can be stabilized for numerical iteration, and overlaps between two such states and one-body Green's functions can be computed. A constrained-path or phaseless approximation can be applied to the random walks of the HFB states if a sign problem or phase problem is present. The method is illustrated with an exact numerical projection in the Kitaev model, and in the Hubbard model with attractive interaction under an external pairing field.

Photoemission spectra from reduced density matrices: The band gap in strongly correlated systems

We present a method for the calculation of photoemission spectra in terms of reduced density matrices. We start from the spectral representation of the one-body Green's function $G$, whose imaginary part is related to photoemission spectra, and we introduce a frequency-dependent effective energy that accounts for all the poles of $G$. Simple approximations to this effective energy give accurate spectra in model systems in the weak as well as strong correlation regime. In real systems reduced density matrices can be obtained from reduced density-matrix functional theory. Here we use this approach to calculate the photoemission spectrum of bulk NiO: our method yields a qualitatively correct picture both in the antiferromagnetic and paramagnetic phases, contrary to mean-field methods, in which the paramagnet is a metal.

Dynamical effects in electron spectroscopy.

One of the big challenges of theoretical condensed-matter physics is the description, understanding, and prediction of the effects of the Coulomb interaction on materials properties. In electronic spectra, the Coulomb interaction causes a renormalization of energies and change of spectral weight. Most importantly, it can lead to new structures, often called satellites. These can be linked to the coupling of excitations, also termed dynamical effects. State-of-the-art methods in the framework of many-body perturbation theory, in particular, the widely used GW approximation, often fail to describe satellite spectra. Instead, approaches based on a picture of electron-boson coupling such as the cumulant expansion are promising for the description of plasmon satellites. In this work, we give a unified derivation of the GW approximation and the cumulant expansion for the one-body Green's function. Using the example of bulk sodium, we compare the resulting spectral functions both in the valence and in the core region, and we discuss the dispersion of quasi-particles and satellites. We show that self-consistency is crucial to obtain meaningful results, in particular, at large binding energies. Very good agreement with experiment is obtained when the intrinsic spectral function is corrected for extrinsic and interference effects. Finally, we sketch how one can approach the problem in the case of the two-body Green's function, and we discuss the cancellation of various dynamical effects that occur in that case.

Electron-hole interactions in correlated electron materials: Optical properties of vanadium dioxide from first principles

Correlated materials have been studied extensively using photoemission spectroscopy. Their optical properties are instead much less explored. Here we present calculations of the optical absorption spectrum of vanadium dioxide $({\mathrm{VO}}_{2})$ in the framework of the Bethe-Salpeter equation (BSE) of many-body perturbation theory. In order to deal with localized electrons we go beyond the standard BSE implementation and extend it to correlated insulators. We show that it is not enough to describe the spectra on the basis of independent electron-hole pairs, even when the electron and hole are separately well described by state-of-the-art one-body Green's functions. Crystal local-field effects are crucial to explain the experimental findings, even qualitatively, and excitonic effects strongly modify the spectra, especially at their onset. In this context, as highighted by the analysis of the BSE results, the quasi-one-dimensional nature of the vanadium-dimer chains plays a prominent role.

Competing phases, phase separation, and coexistence in the extended one-dimensional bosonic Hubbard model

We study the phase diagram of the one-dimensional bosonic Hubbard model with contact $(U)$ and near neighbor $(V)$ interactions focusing on the gapped Haldane insulating (HI) phase which is characterized by an exotic nonlocal order parameter. The parameter regime $(U,V$, and $\ensuremath{\mu})$ where this phase exists and how it competes with other phases, such as the supersolid (SS) phase, is incompletely understood. We use the stochastic Green function quantum Monte Carlo algorithm as well as the density matrix renormalization group to map out the phase diagram. Our main conclusions are that the HI exists only at $\ensuremath{\rho}=1$, the SS phase exists for a very wide range of parameters (including commensurate fillings), and displays power law decay in the one-body Green function. In addition, we show that at fixed integer density, the system exhibits phase separation in the $(U,V)$ plane.

Ab initioself-consistent Gorkov-Green's function calculations of semi-magic nuclei: Numerical implementation at second order with a two-nucleon interaction

Background: The newly developed self-consistent Gorkov-Green's function approach represents a promising path to the ab initio description of mid-mass open-shell nuclei. The formalism based on a two-nucleon interaction and the second-order truncation of Gorkov's self-energy has been described in detail in Ref. [Som\`a, Duguet, and Barbieri, Phys. Rev. C 84, 064317 (2011)].Purpose: The objective is to discuss the methodology used to solve Gorkov's equation numerically and to gauge its performance in view of carrying out systematic calculations of medium-mass nuclei in the future. In doing so, different sources of theoretical error and degrees of self-consistency are investigated.Methods: We employ Krylov projection techniques with a multi-pivot Lanczos algorithm to efficiently handle the growth of poles in the one-body Green's function that arises as a result of solving Gorkov's equation self-consistently. We first characterize the numerical scaling of Gorkov's calculations based on full self-consistency and on a partially self-consistent scheme coined as ``sc0''. Using small model spaces, the Krylov projection technique is then benchmarked against exact diagonalization of the original Gorkov matrix. Next, the convergence of the results as a function of the number ${N}_{\ensuremath{\ell}}$ of Lanczos iterations per pivot is investigated in large model spaces. Eventually, the convergence of the calculations with the size of the harmonic oscillator model space is examined.Results: Gorkov self-consistent Green's function (SCGF) calculations performed on the basis of Krylov projection techniques display a favorable numerical scaling that authorizes systematic calculations of mid-mass nuclei. The Krylov projection selects efficiently the appropriate degrees of freedom while spanning a very small fraction of the original space. For typical large-scale calculations of mid-mass nuclei, a Krylov projection making use of ${N}_{\ensuremath{\ell}}\ensuremath{\approx}50$ yields a sufficient degree of accuracy on the observables of interest. The partially self-consistent sc0 scheme is shown to reproduce fully self-consistent solutions in small model spaces at the 1$%$ level. Eventually, Gorkov-Green's function calculations performed on the basis of SRG-evolved interactions show a fast convergence as a function of the model-space size.Conclusions: The end result is a tractable, accurate and gently scaling ab initio scheme applicable to complete isotopic and isotonic chains in the medium-mass region. The partially self-consistent sc0 scheme provides an excellent compromise between accuracy and computational feasibility and will be the workhorse of systematic Gorkov-Green's function calculations in the future. The numerical scaling and performances of the algorithm employed offers the possibility (i) to apply the method to even heavier systems than those (e.g., ${}^{74}$Ni) already studied so far and (ii) to perform converged Gorkov SCGF calculations based on harder, e.g. original chiral interactions.

Valence Electron Photoemission Spectrum of Semiconductors:Ab InitioDescription of Multiple Satellites

The experimental valence band photoemission spectrum of semiconductors exhibits multiple satellites that cannot be described by the GW approximation for the self-energy in the framework of many-body perturbation theory. Taking silicon as a prototypical example, we compare experimental high energy photoemission spectra with GW calculations and analyze the origin of the GW failure. We then propose an approximation to the functional differential equation that determines the exact one-body Green's function, whose solution has an exponential form. This yields a calculated spectrum, including cross sections, secondary electrons, and an estimate for extrinsic and interference effects, in excellent agreement with experiment. Our result can be recast as a dynamical vertex correction beyond GW, giving hints for further developments.

Electron correlation effects on Lehmann spectra of one-body Green functions for insulating states caused by the Coulomb repulsion

Electron correlation effects on the Lehmann spectra of the one-body Green functions are clarified by using the quantum Monte Carlo simulation of path-integral form. Through the analysis of the Lehmann spectrum, we also show that the electron states of the insulating systems caused by the Coulomb repulsion (Mott-type insulators) significantly depend on the strength of the Coulomb interaction. Specifically, it is shown that in the weak interaction regime, the momentum-specified Lehmann spectrum is qualitatively explained by the second order perturbation theory from the unrestricted Hartree-Fock state, especially near the Fermi energy. This means that the Lehmann spectrum still has a dominant one-body component. On the other hand, in the strong interaction regime, the Lehmann spectrum loses such a one-body component almost completely even near the Fermi energy, and is constituted of the many-body component caused by the coupling between the charge and magnetic excitations. These results indicate that in the weak interaction regime, the electron state of the Mott-type insulator can be described by the one-body picture near the Fermi energy, while in the strong interaction regime, such a one-body picture no longer works even near the Fermi energy. For these two cases, the Lehmann spectra have clearly different peak structures. Since the Lehmann spectrum corresponds to the photoemission spectrum in experiment, we conclude that photoemission spectroscopy can be a valuable tool to obtain direct information on the strength of the Coulomb interaction.

Ring-diagram summations and the self-energy of the homogeneous electron gas at its weak-correlation limit

The small-${r}_{s}$ asymptotics of the self-energy $\ensuremath{\Sigma}(k,\ensuremath{\omega})$ of the homogeneous electron gas (HEG) is studied in terms of the Feynman diagrams involving the noninteracting one-body Green's function ${G}_{0}$ and the static bare Coulomb repulsion ${v}_{0}$. The lowest-order approximation to $\ensuremath{\Sigma}(k,\ensuremath{\omega})$ is given by the product of ${G}_{0}$ and ${v}_{0}$. The nature of the proper ring-diagram summation (equivalent to the random-phase approximation) for $\ensuremath{\Sigma}(k,\ensuremath{\omega})$ that affords the correct small-${r}_{s}$ single behavior of ${r}_{s}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{r}_{s}$ is investigated. Reexamination of ring-diagram summations for several properties of the HEG proves in a rigorous manner that the product ${G}_{0}{v}_{\mathrm{r}}$, where ${v}_{\mathrm{r}}$ is the ring-diagram-summed dynamically screened repulsion, yields the correct lowest-order asymptotics, whereas ${G}_{\mathrm{r}}{v}_{0}$, where ${G}_{\mathrm{r}}$ is the ring-diagram-summed Green's function, contributes only to higher-order terms.

Path-integral theory for photoemission spectra of intermediately correlated systems

By using the quantum Monte-Carlo method, we calculated the Lehmann spectra of momentum-specified one-body Green's functions as well as the total density of states (DOS), for one-dimensional half-filled Hubbard systems with various correlation strengths and temperatures. In the region of the intermediate correlation strength and temperature, we found that the DOS has new characteristic peaks near the Fermi level, in addition to the well-known upper and lower Hubbard-band peaks. We show these new peaks near the Fermi level are quasi-coherent, while the upper and lower Hubbard-band peaks are strongly incoherent. This multi-peaked structure in the intermediate region is also found to be insensitive to the dimensionality. Our results qualitatively agree with the recent experiment on CaVO 3 and SrVO 3.

One-body green functions of half-filled Hubbard models: predominance of multimagnon incoherent component and minuteness of zeromagnon coherent one

Abstract One-body Green functions of one- and two-dimensional half-filled Hubbard models are calculated by means of a quantum Monte Carlo method. To distinguish a zeromagnon coherent component from a multimagnon incoherent one in the Lehmann spectrum of the one-body Green function, a site-diagonal staggered potential Δ is artificially added to the Hubbard model. Due to this Δ, the energy gap opens in the magnon excitation and the zeromagnon coherent component is well separated from the multimagnon incoherent one. Systematic analysis on the Δ-dependence of the coherent and incoherent components clearly shows that the Lehmann spectrum of the one-body Green function is predominated by the incoherent component and the coherent component makes only a minute contribution to the spectrum. Our results indicate that a peak in the angle-resolved photoemission spectrum of the so-called Mott–Hubbard insulator does not correspond to a one-body state.

One-body Green function of half-filled Hubbard model predominated by multi-magnon incoherent component

We show that for one- and two-dimensional half-filled Hubbard models, multi-magnon incoherent component predominates the Lehmann spectrum of the one-body Green function, while zero-magnon coherent component only makes a minute contribution. To distinguish the coherent component from the incoherent one, we add a site-diagonal staggered potential Δ to the usual Hubbard model. This Δ artificially opens a finite energy gap in a magnon excitation, and as a result, the zero-magnon coherent component is clearly separated from the multi-magnon incoherent one. Systematic analysis on Δ-dependence of these two components shows that the weight of the coherent component is less than a few percent in the limit of Δ→0. Our results indicate that a peak in the angle-resolved photoemission spectrum of an insulator driven by the strong Coulomb repulsion originates mainly from this incoherent component. It does not reflect a one-body state.

Minute zero–magnon component and predominant multi-magnon one in photoemission spectra of the two-dimensional half-filled Hubbard system

We show how the coherent and the incoherent components contribute to the one-body Green's function for the two-dimensional half-filled Hubbard model. By adding a site-diagonal staggered potential, a magnon gap artificially opens and, as a result, the zero–magnon coherent component is clearly separated from the multi-magnon incoherent one. We show that the incoherent component predominates the spectrum of the usual Hubbard system, while the coherent component makes only a minute contribution.

On the trees of quantum fields

The solution of some equations involving functional derivatives is given as a series indexed by planar binary trees. The terms of the series are given by an explicit recursive formula. Some algebraic properties of these series are investigated. Several examples are treated in the case of quantum electrodynamics: the complete fermion and photon propagators, the two-body Green function, and the one-body Green function in the presence of an external source, the complete vacuum polarization, electron self-energy and irreducible vertex.