Self-Consistent Green's Function Theory for Atomic Nuclei

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.

The single-particle spectral functions in asymmetric nuclear matter are\ncomputed using the ladder approximation within the theory of finite temperature\nGreen's functions. The internal energy and the momentum distributions of\nprotons and neutrons are studied as a function of the density and the asymmetry\nof the system. The proton states are more strongly depleted when the asymmetry\nincreases while the occupation of the neutron states is enhanced as compared to\nthe symmetric case. The self-consistent Green's function approach leads to\nslightly smaller energies as compared to the Brueckner Hartree Fock approach.\nThis effect increases with density and thereby modifies the saturation density\nand leads to smaller symmetry energies.\n

The strength of the $N=28$ magic number in neutron-rich argon isotopes is examined through high-precision mass measurements of $^{46\text{--}48}\mathrm{Ar}$, performed with the ISOLTRAP mass spectrometer at ISOLDE/CERN. The new mass values are up to 90 times more precise than previous measurements. While they suggest the persistence of the $N=28$ shell closure for argon, we show that this conclusion has to be nuanced in light of the wealth of spectroscopic data and theoretical investigations performed with the SDPF-U phenomenological shell model interaction. Our results are also compared with ab initio calculations using the valence space in-medium similarity renormalization group and the self-consistent Green's function approaches. Both calculations provide a very good account of mass systematics at and around $Z=18$ and, generally, a consistent description of the physics in this region. This combined analysis indicates that $^{46}\mathrm{Ar}$ is the transition between the closed-shell $^{48}\mathrm{Ca}$ and collective $^{44}\mathrm{S}$.

We review some applications of self-consistent Green's function theory to studies of one- and two-nucleon structure in finite nuclei. Large-scale microscopic calculations that employ realistic nuclear forces are now possible. Effects of long-range correlations are seen to play a dominant role in determining the quenching of absolute spectroscopic factors. They also enhance considerably (e,e' pn ) cross sections in superparallel kinematics, in agreement with observations.

An efficient self-consistent Green's-function technique using a generalization of the coherent-potential approximation method is presented in order to describe the electronic structure of inhomogeneous semi-infinite alloys with varying concentration profiles at the surface within the local-density approximation. The formalism is applied to the study of the electronic properties of the (001) surface of ${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ni}}_{\mathit{x}}$ fcc random alloys.

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.

Electronic properties of random surfaces

We present calculations of the energy per particle of pure neutron and symmetric nuclear matter with simplified Argonne nucleon-nucleon potentials for different many-body theories. We compare critically the Brueckner-Hartree-Fock results to other formalisms, such as the Brueckner-Bethe-Goldstone expansion up to third order, self-consistent Green's functions, auxiliary field diffusion Monte Carlo, and Fermi hypernetted chain. We evaluate the importance of spin-orbit and tensor correlations in the equation of state and find these to be important in a wide range of densities.

This work analyses how impurities and vacancies on the surface of a graphene sample affect its optical conductivity and plasmon excitations. The disorder is analysed in the self-consistent Green's function formulation and nonlocal effects are fully taken into account. It is shown that impurities modify the linear spectrum and give rise to an impurity band whose position and width depend on the two parameters of our model, the density and the strength of impurities. The presence of the impurity band strongly influences the electromagnetic response and the plasmon losses. Furthermore, we discuss how the impurity-band position can be obtained experimentally from the plasmon dispersion relation and discuss this in the context of sensing.

Linear optical properties can be accurately calculated using the Bethe-Salpeter equation. After introducing a suitable product basis for the electron-hole pairs, the Bethe-Salpeter equation is usually recast into a complex non-Hermitian eigenvalue problem that is difficult to solve using standard eigenvalue solvers. In solid-state physics, it is therefore common practice to neglect the problematic coupling between the positive- and negative-frequency branches, reducing the problem to a Hermitian eigenvalue problem [Tamm-Dancoff approximation (TDA)]. We use time-inversion symmetry to recast the full problem into a quadratic Hermitian eigenvalue problem, which can be solved routinely using standard eigenvalue solvers even at a finite wave vector $\mathbf{q}$. This allows us to access the importance of the coupling between the positive- and negative-frequency branch for prototypical solids. As a starting point for the Bethe-Salpeter calculations, we use self-consistent Green's-function methods ($\mathit{GW}$), making the present scheme entirely ab initio. We calculate the optical spectra of carbon (C), silicon (Si), lithium fluoride (LiF), and the cyclic dimer ${\mathrm{Li}}_{2}{\mathrm{F}}_{2}$ and discuss why the differences between the TDA and the full solution are tiny. However, at finite momentum transfer $\mathbf{q}$, significant differences between the TDA and our exact treatment are found. The origin of these differences is explained.

We calculate thermodynamic quantities in symmetric nuclear matter within the self-consistent Green's functions method including three-body forces. The thermodynamic potential is computed directly from a diagrammatic expansion, implemented with the CD-Bonn and Nijmegen nucleon-nucleon potentials and the Urbana three-body forces. We present results for entropy and pressure up to temperatures of 20 MeV and densities of 0.32 fm^-3. While the pressure is sensitive to the inclusion of three-body forces, the entropy is not. The unstable spinodal region is identified and the critical temperature associated to the liquid-gas phase transition is determined. When three-body forces are added we find a strong reduction of the critical temperature, obtaining T_c ~ 12 MeV.

Orthogonality instability in a planar antiferromagnet with one impurity

Pairing properties of infinite matter are surveyed that are obtained from self-consistent Green's function calculations. A complete treatment of off-shell propagation that incorporates the effects of short-range correlations is included. Mean-field calculations based on the BCS approach are superseded by the present results, which resolve a long-standing puzzle associated with 3S1-3D1 (proton-neutron) pairing in symmetric nuclear matter. Results for 1S0 pairing in pure neutron matter are in agreement with recent Monte-Carlo calculations for this system. The possibility of proton pairing in finite nuclei, as a result of increasing proton correlations with increasing nucleon asymmetry, is pointed out.

ABSTRACTA new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm−1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.

The eigenfunction expansion of the infinite depth free surface Green function in three dimensions