Green's functions and response functions in geochemical modelling

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.

Herein, a highly practical first‐principles approach for studying phonon transport across various materials, ranging from low‐dimensional systems to three‐dimensional bulk structures, as well as nanostructures, is presented. This method integrates the nonequilibrium Green's function (NEGF) formalism with ab initio calculations of dynamical matrices generated by Quantum Espresso. A detailed technical approach for extracting atomistic force constants and forming dynamical matrices that adhere to the tri‐diagonal technique within the NEGF framework is provided. Additionally, an efficient method for constructing dynamical matrices of conventional cells from those of primary cells is introduced, remarkably reducing the computational costs associated with density‐functional theory calculations. The robustness and applicability of this methodology are validated through several case studies.

This paper describes the point defect treatment in nanowire MOSFETs. The 3D Poisson equation is self-consistently coupled to the Schrodinger equation. We adopt the mode-space representation in which the 3D Schrodinger equation is divided into a 2D equation, including the confinement of the cross-section, and an ID equation describing the ballistic transport based on the nonequilibrium Green's function formalism (NEGF). We use the simplest form of the tight-binding theory, with one orbital per atom, and considering only the first-neighbor-interactions. In this framework, equivalent to the ellipsoidal energy band approximation, influence of point defects on electronic transport is discussed.

The electronic structure of transition metal tricarbonyl derivatives in the ground state and the cationic hole-states. A semiempirical indo mo investigation based on the Green's function formalism

Several ultrasound-based techniques for the estimation of soft tissue elasticity are currently being investigated. Most of them study the medium response to dynamic excitations. Such responses are usually modeled in a purely elastic medium using a Green's function solution of the motion equation. However, elasticity by itself is not necessarily a discriminant parameter for malignancy diagnosis. Modeling viscous properties of tissues could also be of great interest for tumor characterization. We report in this paper an explicit derivation of the Green's function in a viscous and elastic medium taking into account shear, bulk, and coupling waves. From this theoretical calculation, 3D simulations of mechanical waves in viscoelastic soft tissues are presented. The relevance of the viscoelastic Green's function is validated by comparing simulations with experimental data. The experiments were conducted using the supersonic shear imaging (SSI) technique which dynamically and remotely excites tissues using acoustic radiation force. We show that transient shear waves generated with SSI are modeled very precisely by the Green's function formalism. The combined influences of out-of-plane diffraction, beam shape, and shear viscosity on the shape of transient waves are carefully studied as they represent a major issue in ultrasound-based viscoelasticity imaging techniques.

The electron-gas problem is investigated by means of a self-consistent Green's-function formalism with the aim of developing practical approximation schemes for metallic densities. The work is based on the Dyson equation for the single-particle propagator $G[U]$ as a functional of an external potential $U$. The self-energy functional $\ensuremath{\Sigma}[U]$, appearing in the Dyson equation, is evaluated by perturbation theory in terms of the exact $G[U]$, thereby leading to a self-consistent problem. A hierarchy of approximations is generated by summing successively larger sets of graphs for $\ensuremath{\Sigma}[U]$. The Dyson equation is expanded in a functional Taylor series in $U$ and yields a nonlinear integral equation for the $U=0$ propagator as well as linear integral equations for the $U=0$ higher-order Green's functions, with kernels dependent on $\frac{\ensuremath{\delta}\ensuremath{\Sigma}}{\ensuremath{\delta}U}$. In applications of the theory, the emphasis is on calculating the longitudinal dielectric function $\ensuremath{\epsilon}$ in terms of the contracted four-point Green's function. The linear integral equation for the latter is solved after making a low-momentum dominance approximation to the kernel. The result is a general, but approximate, closed-form expression for $\ensuremath{\epsilon}$ which can be used for different choices of $\ensuremath{\Sigma}$. The following five approximations for $\ensuremath{\epsilon}$, based on different approximations for $\ensuremath{\Sigma}$, are presented: the Hartree-Fock, random-phase, generalized random-phase, second-stage random-phase, and low-high-density approximations. The last approximation is designed to work well at the two extremes of the density spectrum and, hopefully, also at metallic densities. The long-wavelength plasmon dispersion relations obtained from two different versions of the generalized random-phase approximation for $\ensuremath{\epsilon}$ agree closely with the results of Kanazawa et al. and of Singwi et al.

We extend the self-consistent Green's functions formalism to take into account three-body interactions. We analyze the perturbative expansion in terms of Feynman diagrams and define effective one- and two-body interactions, which allows for a substantial reduction of the number of diagrams. The procedure can be taken as a generalization of the normal ordering of the Hamiltonian to fully correlated density matrices. We give examples up to third order in perturbation theory. To define nonperturbative approximations, we extend the equation of motion method in the presence of three-body interactions. We propose schemes that can provide nonperturbative resummation of three-body interactions. We also discuss two different extensions of the Koltun sum rule to compute the ground state of a many-body system.

In this work we analyze the size (radius) distribution function of dust particles in tokamak plasmas during a steady state discharge. A relation between the radius distribution function of dust in the plasma and the radius distribution of dust injected from tokamak walls is obtained using a Green's function formalism. Numerical simulations of the dust radius distribution function in a tokamak plasma with the Dust Transport (DUSTT) code are used to obtain the analytical form of the Green's function semi‐empirically. It is demonstrated that the Green's function obtained can be used to predict qualitatively the dust size distributions in the tokamak plasmas. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Korringa-Kohn-Rostoker (KKR) Green's function, multiple-scattering theory is an efficient site-centered, electronic-structure technique for addressing an assembly of $N$ scatterers. Wave-functions are expanded in a spherical-wave basis on each scattering center and indexed up to a maximum orbital and azimuthal number $L_{max}=(l,m)_{max}$, while scattering matrices, which determine spectral properties, are truncated at $L_{tr}=(l,m)_{tr}$ where phase shifts $\delta_{l>l_{tr}}$ are negligible. Historically, $L_{max}$ is set equal to $L_{tr}$; however, a more proper procedure retains free-electron and single-site contributions for $L_{max}>L_{tr}$ with $\delta_{l>l_{tr}}$ set to zero [Zhang and Butler, Phys. Rev. B {\bf 46}, 7433]. We present a numerically efficient and accurate \emph{augmented}-KKR Green's function formalism that solves the KKR secular equations by matrix inversion [$\mathcal{R}^3$ process with rank $N(l_{tr}+1)^2$] and includes higher-order $L$ contributions via linear algebra [$\mathcal{R}^2$ process with rank $N(l_{max}+1)^2$]. Augmented-KKR yields properly normalized wave-functions, numerically cheaper basis-set convergence, and a total charge density and electron count that agrees with Lloyd's formula. For fcc Cu, bcc Fe and L$1_0$ CoPt, we present the formalism and numerical results for accuracy and for the convergence of the total energies, Fermi energies, and magnetic moments versus $L_{max}$ for a given $L_{tr}$.

The Green's function formalism in Condensed Matter Physics is reviewed within the equation of motion approach. Composite operators and their Green's functions naturally appear as building blocks of generalized perturbative approaches and require fully self-consistent treatments in order to be properly handled. It is shown how to unambiguously set the representation of the Hilbert space by fixing both the unknown parameters, which appear in the linearized equations of motion and in the spectral weights of non-canonical operators, and the zero-frequency components of Green's functions in a way that algebra and symmetries are preserved. To illustrate this procedure some examples are given: the complete solution of the two-site Hubbard model, the evaluation of spin and charge correlators for a narrow-band Bloch system, the complete solution of the three-site Heisenberg model, and a study of the spin dynamics in the Double-Exchange model.

Superconducting electronics are promising candidates for complementing semiconductor electronics. However, a comprehensive suite of superconducting design and simulation tools that is necessary to streamline and automate the design process is either nonexistent or very limited. Here, we describe Synopsys' advances in the technology computer-aided design (TCAD) front for physics-based simulation of superconducting electronics based on Josephson junctions (JJs). We employ Keldysh-Gor'kov-Nambu Green's function formalism within the context of the self-consistent mean-field theory of Bogoliubov and de-Gennes to simulate the equilibrium and transport properties of JJ-based devices in one dimension. Using these Green's functions, we calculate various electrical properties and effects, including order parameter and proximity effect, current-phase characteristics and current-density distribution for arbitrary temperatures, material properties and composition, and device dimensions. We further expand the TCAD simulator using the Floquet-Keldysh-Gor'kov-Nambu Green's function for calculating current-voltage characteristics of JJs in nonequilibrium and capture various quantum mechanical effects including multiple Andreev reflections (MAR) for constant applied voltage.

We develop a Green's function formalism and with it we are able to obtain close expressions for the tunnelling and for the reflection times. For each problem, there appear two characteristic times which correspond to the real and imaginary parts of the integral of the Green's function at coinciding coordinates. The time related to the imaginary part represents the minimum uncertainty of the measurement. A strong analogy between the results of the different existing approaches is established, and we show that their main differences are due to finite-size effects.

Retarded green's-function calculation of magnetic resonance relaxation time T1 for rotating molecules or atomic groups in solids

In quantum statistical mechanics, the Green's function formalism provides an expression for the density of a fluid as a four-dimensional momentum-energy integral over the spectral function. This function can be expressed in terms of the complex self-energy of the single-particle excited states. By using the “ladder diagram” approximation, in a low activity limit at which Fermi-Dirac and Bose-Einstein distributions can be approximated by a Boltzmann distribution, the self-energy has been expressed in terms of the two-body scattering amplitude. Density and pressure can then be expressed in terms of the activity, the temperature, and the two-body scattering phase shifts. A complete numerical evaluation of these results has been made for the case of argon at 100‡K, represented by a hard-sphere plus square-well potential: results are presented for the complex self-energy, the density, and the pressure as a function of activity. The resulting equation of state is compared to experimental results represented by the Beattie-Bridgeman equation and good agreement is found for the gaseous part of the 100‡K isotherm. Furthermore, two simple analytic equations of state are derived from these expressions with additional (low-density) approximations, which resemble closely some of the equations obtained from the lattice gas theories.

A magnetic superlattice with three‐layer (plane) unit cell was treated within the boson formalism. The energies were derived using Green's functions formalism, and it is shown that previous calculations did not take into account the principal demand that excitation energies should be positive, so many previous conclusions should be re‐examined. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)