Quasiparticle Calculations in Solids

The green's function method in quantum statistics

The GW approximation (GWA) extends the well-known Hartree-Fock approximation (HFA) for the self-energy (exchange potential), by replacing the bare Coulomb potential v by the dynamically screened potential W, e.g. Vex = iGv is replaced by GW = iGW. Here G is the one-electron Green's function. The GWA like the HFA is self-consistent, which allows for solutions beyond perturbation theory, like say spin-density waves. In a first approximation, iGW is a sum of a statically screened exchange potential plus a Coulomb hole (equal to the electrostatic energy associated with the charge pushed away around a given electron). The Coulomb hole part is larger in magnitude, but the two parts give comparable contributions to the dispersion of the quasi-particle energy. The GWA can be said to describe an electronic polaron (an electron surrounded by an electronic polarization cloud), which has great similarities to the ordinary polaron (an electron surrounded by a cloud of phonons). The dynamical screening adds new crucial features beyond the HFA. With the GWA not only bandstructures but also spectral functions can be calculated, as well as charge densities, momentum distributions, and total energies. We will discuss the ideas behind the GWA, and generalizations which are necessary to improve on the rather poor GWA satellite structures in the spectral functions. We will further extend the GWA approach to fully describe spectroscopies like photoemission, x-ray absorption, and electron scattering. Finally we will comment on the relation between the GWA and theories for strongly correlated electronic systems. In collecting the material for this review, a number of new results and perspectives became apparent, which have not been published elsewhere.

Ab initio green's function calculations on highly conducting polymers: Effects of electron correlation and aperiodicity

Herein, we present a fragment-based approach for calculating the charged and neutral excited states in molecular systems, based on the many-body Green's function method within the GW approximation and the Bethe-Salpeter equation (BSE). The implementation relies on the many-body expansion of the total irreducible polarizability on the basis of fragment molecular orbitals. The GW quasi-particle energies in complex molecular environments are obtained by the GW calculation for the target fragment plus induced polarization contributions of the surrounding fragments at the static Coulomb-hole plus screened exchange level. In addition, we develop a large-scale GW/BSE method for calculating the delocalized excited states of molecular aggregates, based on the fragment molecular orbital method and the exciton model. The accuracy of fragment-based GW and GW/BSE methods was evaluated on molecular clusters and molecular crystals. We found that the accuracy of the total irreducible polarizability can be improved systematically by including two-body correction terms, and the fragment-based calculations can reasonably reproduce the results of the corresponding unfragmented calculations with a relative error of less than 100 meV. The proposed approach enables efficient excited-state calculations for large molecular systems with reasonable accuracy.

It was recently proposed to use variational functionals based on many‐body perturbation theory for the calculation of the total energies of many‐electron systems. The accuracy of such functionals depends on the degree of sophistication of the underlying perturbation expansions. The energy functionals are variational in the sense that they can be evaluated at rather crude approximations to their independent variables, which are the one‐electron Green function, or the one‐electron Green function and the dynamically screened electron interaction. The functionals were previously applied to the electron gas and shown to be extraordinarily accurate already at the level of the so‐called GW approximation (GWA). In the current work we have tested the functional due to Luttinger and Ward, which is a functional of the Green function. Using density functional theory (DFT) and Hartree–Fock Green functions as input variables, we have calculated total energies of diatomic molecules at the level of the GWA as well as with second‐order exchange effects included. We will also discuss various other variational energy functionals, including DFT orbital functionals based on many‐body perturbation theory. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

ABSTRACTMany‐body Green's function theory in the GW approximation with the Bethe–Salpeter equation (BSE) provides a powerful framework for the first‐principles calculations of single‐particle and electron–hole excitations in perfect crystals and molecules alike. Application to complex molecular systems, for example, solvated dyes, molecular aggregates, thin films, interfaces, or macromolecules, is particularly challenging as they contain a prohibitively large number of atoms. Exploiting the often localized nature of excitation in such disordered systems, several methods have recently been developed in which GW‐BSE is applied to a smaller, tractable region of interest that is embedded into an environment described with a lower‐level method. Here, we review the various strategies proposed for such embedded many‐body Green's functions approaches, including quantum–quantum and quantum–classical embeddings, and focus in particular on how they include environment screening effects either intrinsically in the screened Coulomb interaction in the GW and BSE steps or via extrinsic electrostatic couplings.

Investigation of the complete valence shell of formic acid by electron momentum spectroscopy and Green's function methods

<sec><p indent="0mm">Exotic nuclei far from the β stability line have become important scientific goals in the studies of experimental physics at large scientific facilities and theoretical research due to the rich new physics. In these exotic nuclei, the neutron or proton Fermi level is close to the continuum threshold. The pairing correlation could scatter the valence neutrons or protons into the continuum. This leads to the extended neutron or proton density distributions in these exotic nuclei. Therefore, properly describing the pairing correlation and continuum is crucial for studying the structures and properties of these exotic nuclei. The Hartree-Fock-Bogoliubov (HFB) theory is one of the promising tools for describing exotic nuclei. In this theory, people usually solve the HFB equation directly in the coordinate space or the Woods-Saxon basis. In the coordinate space, when employing box boundary conditions to discretize the continuum, the behavior of the wave functions at the boundaries will be affected by the size of the box. Meanwhile, one could not obtain the energy and width of the resonant states directly from the discretized continuum states. The Green’s function (GF) is a simple and effective tool for handling the continuum, which has been widely used in nuclear structure research. The GF can be established by using the wave functions that satisfy the equation of motion and the proper bound boundary conditions for the bound and continuum states. Then one can use the loop integral of the GF on the complex energy plane to construct the particle density and level density. In this way, the continuum states can be included in the density with proper boundary conditions, and thus the extended density distribution can be properly described. Furthermore, the GF on the complex energy plane and the level density can be used to identify the resonant states directly. </sec><sec> This paper briefly reviews the development of the GF method in the continuum density functional theory, focusing on its application in describing halo phenomena and single-particle resonance states in exotic nuclei. For the description of the halo phenomena, this review takes the neutron-rich Zr as examples to show the results given by Skyrme HFB calculations with the GF method. The extended neutron particle and pair density distributions can be more properly described by the GF method, compared to the box-discretized method. The self-consistency to deal with the pairing correlation and the continuum is important to determine the asymptotic neutron particle and pair density distributions. Furthermore, with the level density obtained by the loop integral of the GF on the complex energy plane, one can describe both the bound and resonant states on the same footing. Recently, in the relativistic mean field theory, a new method to obtain precise information for the resonant energy and width is proposed by using directly the poles of the GF on the complex energy plane. This helps explain the conservation and breaking of pseudospin symmetry in the nucleon single-particle levels. Additionally, Green’s function method is easily compatible with various theoretical models, thus in the future it can be further applied to describe the exotic nuclear collective resonances and nuclear reaction processes. </sec>

A theoretical model for evaluating solid multilayered spherical solids heated by a frequency-modulated light beam using the Green function method is presented. The specific thermal-wave Green function corresponding to the composite structure has been derived. The characteristics of the thermal-wave field with respect to the thermophysical, geometrical, and measurement parameters are presented. Unlike the quadrupole method, the Green function method is capable of evaluating thermal-wave fields at any point of multilayered structures with arbitrary intensity distributions of the incident laser beams. This study establishes applications of thermal-wave fields in both cylindrical and spherical samples using the Green function method and is of importance in characterizing radially inhomogeneous spherical solids.

The electronic structures, absorption spectra and colors of Cu are calculated. Calculations are performed in the GW approximation (GWA) approximation, where G refers to Green's function and W is the dynamically screened Coulomb interaction. The calculated absorption spectra and color of Cu based on the density functional theory and the GWA are presented, and the calculated results within the GWA agree well with measurements. The calculated results indicate that many-body effects play an important role for the quasi-particle property calculations of Cu.

Topological phase transitions in free fermion systems can be characterized by closing of single-particle gap and change in topological invariants. However, in the presence of electronic interactions, topological phase transitions are more complicated. In paper I of this series (arXiv:1510.07816), we have developed an efficient scheme to evaluate the topological invariants based on Green's function formalism. Here, in paper II, we demonstrate four interaction-drive topological phase transitions (TPTs) in two-dimensional (2D) interacting topological insulators (TIs) via large-scale quantum Monte Carlo (QMC) simulations, based on the scheme of evaluating topological invariants presented in paper I. Across these transitions, the defining symmetries of the TIs have been neither explicitly nor spontaneously broken. In the first two models, the topological invariants calculated from Green's function formalism succeed in characterizing interaction-driven TPTs. However, in the second two models, we find single-particle gap does not close and the topological invariants constructed from single-particle Green's function acquire no change across the TPTs. Unexpected breakdown of the Green's function formalism in constructing topological invariants is thus discovered. We thence classify the TPTs in interacting TIs into two categories: those have noninteracting correspondence can be characterized successfully by the topological invariants constructed from Green's functions, while for the others that do not have noninteracting correspondence, the Green's function formalism experiences a breakdown but more interesting and exciting phenomena, such as emergent collective critical modes at the transition, arise. Discussion on the success and breakdown of topological invariants constructed from the Green's function formalism in the context of symmetry protected topological (SPT) states is presented.

Abstractauthoren We present a first-principles many-body Green's function method (GW approximation and Bethe–Salpeter equation) of the electronic and optical properties of recently predicted hydrogen–graphene–fluorine janus graphene. Significant self-energy corrections, of more than 50%, to the Kohn–Sham bandgap from the local density approximation (LDA) calculations are found. Moreover, the optical absorption spectrum of this janus graphene is dominated by enhanced excitonic effects with formation of a bound exciton with considerable binding energy. The reduced spatial separation of excited electrons and holes gives rise to extremely short radiative lifetimes, preventing condensation.

An effective method to calculate the electrostatic force between a microscopic tip and an infinite metallic plane is proposed and tested. The Green function method is used to integrate out exactly the potential distribution in the plane. The plane plus tip problem is thus reduced to a boundary value problem for the tip surface only, enabling solution by a standard numerical method. The main advantage of this approach is that systems with an arbitrary plane potential distribution may be solved with the same method and comparable numerical efficiency to problems with a constant plane potential. The method is used to calculate the electrostatic force on a tip due to a plane with a step potential and a local quadratic potential island. The calculated perpendicular force components agree with previously published theoretical results. In addition, lateral components of the tip-surface force are computed and shown to be comparable to the vertical component close to a potential step, in agreement with the published experimental data. This suggests the possibility of using lateral, as well as vertical force data in atomic force microscopy for the detection and imaging of potential steps.

We present recent advances in numerical implementations of hybrid functionals and the GW approximation within the full-potential linearized augmented-plane-wave (FLAPW) method. The former is an approximation for the exchange–correlation contribution to the total energy functional in density-functional theory, and the latter is an approximation for the electronic self-energy in the framework of many-body perturbation theory. All implementations employ the mixed product basis, which has evolved into a versatile basis for the products of wave functions, describing the incoming and outgoing states of an electron that is scattered by interacting with another electron. It can thus be used for representing the nonlocal potential in hybrid functionals as well as the screened interaction and related quantities in GW calculations. In particular, the six-dimensional space integrals of the Hamiltonian exchange matrix elements (and exchange self-energy) decompose into sums over vector–matrix–vector products, which can be evaluated easily. The correlation part of the GW self-energy, which contains a time or frequency dependence, is calculated on the imaginary frequency axis with a subsequent analytic continuation to the real axis or, alternatively, by a direct frequency convolution of the Green function G and the dynamically screened Coulomb interaction W along a contour integration path that avoids the poles of the Green function. Hybrid-functional and GW calculations are notoriously computationally expensive. We present a number of tricks that reduce the computational cost considerably, including the use of spatial and time-reversal symmetries, modifications of the mixed product basis with the aim to optimize it for the correlation self-energy and another modification that makes the Coulomb matrix sparse, analytic expansions of the interaction potentials around the point of divergence at k = 0, and a nested density and density-matrix convergence scheme for hybrid-functional calculations. We show CPU timings for prototype semiconductors and illustrative results for GdN and ZnO.

Simulations that are meant to determine the steady-state distribution of a diffusible solute such as oxygen in tissues have typically used finite difference methods to solve the diffusion equation. Finite difference methods require a tissue mesh with enough points to resolve oxygen gradients near and between discrete blood vessels. The large number of points that are typically required can make these calculations very slow. In this paper, we investigate a numerical method known as the Green's function method which is not bound by the same constraint. The Green's function method is expected to yield an accurate oxygen distribution more quickly by requiring fewer mesh points. Both methods were applied to calculate the steady state oxygen distribution in a model simulation region. When the Green's function calculation used meshes with 1/2, 1/4 and, 1/8 of the resolution required for the finite-difference mesh, there was good agreement with the finite difference calculation in all cases. When the volume of the domain was increased 8-fold the Green's function method was able to calculate the O2 field in 22 minutes, whereas the finite difference calculation is expected to take approximately 1 week. The number of steps required for the Green's function calculation increases quadratically with the number of points in the tissue mesh. As a result, small meshes are calculated very quickly using Green's functions, while for larger mesh sizes this method experiences a significant decrease in efficiency.