Second-order Diagrams Research Articles

The Iconic-Symbolic Spectrum

It is common to distinguish two great families of representation. Symbolic representations include logical and mathematical symbols, words, and complex linguistic expressions. Iconic representations include dials, diagrams, maps, pictures, 3-dimensional models, and depictive gestures. This essay describes and motivates a new way of distinguishing iconic from symbolic representation. It locates the difference not in the signs themselves, nor in the contents they express, but in the semantic rules by which signs are associated with contents. The two kinds of rule have divergent forms, occupying opposite poles on a spectrum of naturalness. Symbolic rules are composed entirely of primitive juxtapositions of sign types with contents, while iconic rules determine contents entirely by uniform natural relations with sign types. This distinction is marked explicitly in the formal semantics of familiar sign systems, both for atomic first-order representations, like words, pixel colors, and dials, and for complex second-order representations, like sentences, diagrams, and pictures.

Pauli rearrangement potential for a scattering state with the nucleon-nucleon interaction in chiral effective field theory

The Pauli rearrangement potential given by the second-order diagram is evaluated for a nucleon optical model potential (OMP) with $G$ matrices of the nucleon-nucleon interaction in chiral effective field theory. The results obtained in nuclear matter are applied for $^{40}$Ca in a local-density approximation. The repulsive effect is of the order of 5MeV at the normal density. The density dependence indicates that the real part of the microscopic OMP becomes shallower in a central region, but is barely affected in a surface area. This improves the overall resemblance of the microscopic OMP to the empirical one.

Diagrammatic expansion for positive density-response spectra: Application to the electron gas

In a recent paper [Phys. Rev. B 90, 115134 (2014)] we put forward a diagrammatic expansion for the self-energy which guarantees the positivity of the spectral function. In this work we extend the theory to the density response function. We write the generic diagram for the density-response spectrum as the sum of partitions. In a partition the original diagram is evaluated using time-ordered Green's functions (GF) on the left-half of the diagram, antitime-ordered GF on the right-half of the diagram and lesser or greater GF gluing the two halves. As there exist more than one way to cut a diagram in two halves, to every diagram corresponds more than one partition. We recognize that the most convenient diagrammatic objects for constructing a theory of positive spectra are the half-diagrams. Diagrammatic approximations obtained by summing the squares of half-diagrams do indeed correspond to a combination of partitions which, by construction, yield a positive spectrum. We develop the theory using bare GF and subsequently extend it to dressed GF. We further prove a connection between the positivity of the spectral function and the analytic properties of the polarizability. The general theory is illustrated with several examples and then applied to solve the long-standing problem of including vertex corrections without altering the positivity of the spectrum. In fact already the first-order vertex diagram, relevant to the study of gradient expansion, Friedel oscillations, etc., leads to spectra which are negative in certain frequency domain. We find that the simplest approximation to cure this deficiency is given by the sum of the zero-th order bubble diagram, the first-order vertex diagram and a partition of the second-order ladder diagram. We evaluate this approximation in the 3D homogeneous electron gas and show the positivity of the spectrum for all frequencies and densities.

On the expressiveness of second-order spider diagrams

Existing diagrammatic notations based on Euler diagrams are mostly limited in expressiveness to monadic first-order logic with an order predicate. The most expressive monadic diagrammatic notation is known as spider diagrams of order. A primary contribution of this paper is to develop and formalise a second-order diagrammatic logic, called second-order spider diagrams, extending spider diagrams of order. A motivation for this lies in the limited expressiveness of first-order logics. They are incapable of defining a variety of common properties, like ‘is even’, which are second-order definable. We show that second-order spider diagrams are at least as expressive as monadic second-order logic. This result is proved by giving a method for constructing a second-order spider diagram for any regular expression. Since monadic second-order logic sentences and regular expressions are equivalent in expressive power, this shows second-order spider diagrams can express any sentence of monadic second-order logic.

Λ, Σ, and Ξ hyperons in neutron matter

Abstract Hyperon single-particle potentials are calculated in pure neutron matter in the framework of the lowest-order Brueckner theory, using two recent baryon–baryon interactions, the SU 6 quark-model potential and the potential derived from the chiral effective field theory. These properties are important for understanding neutron star matter on the basis of underlying baryon–baryon interactions. Because the calculated potential of Σ − is strongly repulsive and that of Ξ − is also repulsive, these hyperons are unlikely to appear in neutron star matter. The Λ potential is attractive enough to appear in high neutron matter as has been commonly expected in microscopic calculations. After showing important contributions of three-nucleon forces in neutron matter, analogous repulsive contributions to the Λ potential from the Σ ⁎ excitation are estimated by evaluating second-order diagrams.

Isotope shifts and relativistic shifts of Cr ii for the study ofαvariation in quasar absorption spectra

We use the combination of configuration interaction and many-body perturbation theory method (CI+MBPT) to perform ab initio calculations the low-energy spectra of Cr II with high accuracy. It is found that second-order MBPT diagrams should be included in a consistent and complete way for the MBPT to improve the accuracy of calculations in this five-valence-electron system. This contrasts with previous ions with fewer valence electrons where it was found that single-valence-electron diagrams dominate the corrections. Isotope shifts and relativistic shifts (q-values) are calculated for use in astronomical determination of the fine-structure constant in quasar absorption spectra.

GWapproximations and vertex corrections on the Keldysh time-loop contour: Application for model systems at equilibrium

We study the effects of self-consistency and vertex corrections on different $GW$-based approximations for model systems of interacting electrons. For dealing with the most general case, we use the Keldysh time-loop contour formalism to evaluate the single-particle Green's functions. We provide the formal extension of Hedin's $GW$ equations for the Green's function in the Keldysh formalism. We show an application of our formalism to the plasmon model of a core electron within the plasmon-pole approximation. We study in detail the effects of the diagrammatic perturbation expansion of the core-electron/plasmon coupling on the spectral functions in the so-called S model. The S model provides an exact solution at equilibrium for comparison with the diagrammatic expansion of the interaction. We show that self-consistency is essential in $GW$-based calculations to obtain the full spectral information. The second-order exchange diagram (i.e., a vertex correction) is also crucial to obtain the good spectral description of the plasmon satellites. We corroborate these results by considering conventional equilibrium $GW$-based calculations for the pure jellium model. We find that with no second-order vertex correction, one cannot obtain the full set of plasmon side-band resonances. We also discuss in detail the formal expression of the Dyson equations obtained for the time-ordered Green's function at zero and finite temperature from the Keldysh formalism and from conventional equilibrium many-body perturbation theory.

Nonequilibrium inelastic electronic transport: Polarization effects and vertex corrections to the self-consistent Born approximation

We study the effect of electron-vibron interactions on the inelastic transport properties of single-molecule nanojunctions. We use the non-equilibrium Green's functions technique and a model Hamiltonian to calculate the effects of second-order diagrams (double-exchange DX and dressed-phonon DPH diagrams) on the electron-vibration interaction and consider their effects across the full range of parameter space. The DX diagram, corresponding to a vertex correction, introduces an effective dynamical renormalization of the electron-vibron coupling in both the purely inelastic and the inelastic-resonant features of the IETS. The purely inelastic features correspond to an applied bias around the energy of a vibron, while the inelastic-resonant features correspond to peaks (resonance) in the conductance. The DPH diagram affects only the inelastic resonant features. We also discuss the circumstances in which the second-order diagrams may be approximated in the study of more complex model systems.

Anisotropic quasiparticle lifetimes in Fe-based superconductors

We study the dynamical quasiparticle scattering by spin and charge fluctuations in Fe-based pnictides within a five-orbital model with on-site interactions. The leading contribution to the scattering rate is calculated from the second-order diagrams with the polarization operator calculated in the random-phase approximation. We find one-particle scattering rates which are highly anisotropic on each Fermi surface sheet due to the momentum dependence of the spin susceptibility and the multi-orbital composition of each Fermi pocket. This fact, combined with the anisotropy of the effective mass, produces disparity between electrons and holes in conductivity, the Hall coefficient, and the Raman initial slope, in qualitative agreement with experimental data.

Anomalous transport and phonon renormalization in a chain with transverse and longitudinal vibrations

We study thermal transport in a chain of coupled atoms, which can vibrate in longitudinal as well as transverse directions. The particles interact through anharmonic potentials upto cubic order. The problem is treated quantum mechanically. We first calculate the phonon frequencies self-consistently taking into account the anharmonic interactions. We show that for all the modes, frequencies must have linear dispersion with wave vector q for small q irrespective of their bare dispersions. We then calculate the phonon relaxation rates Γi(q), where i is the polarization index of the mode, in a self-consistent approximation based on second-order perturbation diagrams. We find that the relaxation rate for the longitudinal phonon, Γx(q)∝q(3/2), while that for the transverse phonon Γy(q)∝q2. The consequence of these results on the thermal conductivity κ(N) of a chain of N particles is that κ(N)∝N(1/2).

Nonequilibrium electronic structure of interacting single-molecule nanojunctions: Vertex corrections and polarization effects for the electron-vibron coupling

We consider the interaction between electrons and molecular vibrations in the context of electronic transport in nanoscale devices. We present a method based on nonequilibrium Green's functions to calculate both equilibrium and nonequilibrium electronic properties of a single-molecule junction in the presence of electron-vibron interactions. We apply our method to a model system consisting of a single electronic level coupled to a single vibration mode in the molecule, which is in contact with two electron reservoirs. Higher-order diagrams beyond the usual self-consistent Born approximation (SCBA) are included in the calculations. In this paper we consider the effects of the double-exchange diagram and the diagram in which the vibron propagator is renormalized by one electron-hole bubble. We study in detail the effects of the first- and second-order diagrams on the spectral functions for a large set of parameters and for different transport regimes (resonant and off-resonant cases), both at equilibrium and in the presence of a finite applied bias. We also study the linear response (linear conductance) of the nanojunction for all the different regimes. We find that it is indeed necessary to go beyond the SCBA in order to obtain correct results for a wide range of parameters.

A relativistic structure of one-meson and one-gluon exchange forces and a lower excitation spectrum of the nucleon and Δ

A lower excitation spectrum of the nucleon and Δ is calculated in a relativistic chiral quark model. Corrections to the baryon mass spectrum from the second-order self-energy and exchange diagrams induced by pion and gluon fields are estimated in the field-theoretical framework. Convergent results for the self-energy terms are obtained when including the intermediate quark and antiquark states with a total momentum up to j = 25/2. Relativistic one-meson and color-magnetic one-gluon exchange forces are shown to generate spin 0, 1, 2, etc operators, which couple the lower and the upper components of the two interacting valence quarks and yield reasonable matrix elements for the lower excitation spectrum of the nucleon and Δ. The only contribution to the ground-state nucleon and Δ comes from the spin-1 operators, which correspond to the exchanged pion or gluon in the l = 1 orbit, thus indicating that both pion exchange and color-magnetic gluon exchange forces can contribute to the spin of baryons. Is is shown also that the contribution of the color-electric component of the gluon fields to the baryon spectrum is enormously large (more than 500 MeV with a value of αs = 0.65) and one needs to restrict to very small values of the strong coupling constant or to exclude completely the gluon-loop corrections to the baryon spectrum. With this restriction, the calculated spectrum reproduces the main properties of the data; however it needs further contribution from the two-pion exchange and instanton-induced exchange (for the nucleon sector) forces in consistence with the realistic NN-interaction models.

Real-Time Path Planning in Dynamic Virtual Environments Using Multiagent Navigation Graphs

We present a novel approach for efficient path planning and navigation of multiple virtual agents in complex dynamic scenes. We introduce a new data structure, Multi-agent Navigation Graph (MaNG), which is constructed using first- and second-order Voronoi diagrams. The MaNG is used to perform route planning and proximity computations for each agent in real time. Moreover, we use the path information and proximity relationships for local dynamics computation of each agent by extending a social force model [Helbing05]. We compute the MaNG using graphics hardware and present culling techniques to accelerate the computation. We also address undersampling issues and present techniques to improve the accuracy of our algorithm. Our algorithm is used for real-time multi-agent planning in pursuit-evasion, terrain exploration and crowd simulation scenarios consisting of hundreds of moving agents, each with a distinct goal.

Short-range potentials from QCD at order g2

We systematically compute the effective short-range potentials arising from second-order QCD diagrams related to bound states of quarks, antiquarks and gluons. Our formalism relies on the assumption that the exchanged gluons are massless, while the constituent gluons as well as the lightest quarks acquire a nonvanishing constituent mass because of confinement. We recover the well-known Fermi–Breit interaction in the heavy quark limit. Such an effective potential has been proved in the past to be relevant for the building of accurate potential models describing usual hadrons (mesons and baryons). The general potentials we obtain in this work are also expected to be useful in the understanding of exotic hadrons such as glueballs and hybrids. In particular, we compute for the first time an effective short-range quark–gluon potential, and show the existence of a quadrupolar interaction term in this case. We also discuss the influence of a possible nonzero mass for the exchanged gluons.

Lower excitation spectrum of the nucleon and delta in a relativistic chiral quark model

The lower excitation spectrum of the nucleon and Δ is calculated in a relativistic chiral quark model. Contributions of the second-order self-energy and exchange diagrams due to pion fields to the mass spectrum of the SU(2) baryons are estimated. A splitting between N(939) and positive parity nucleon resonance (Roper resonance) N*(1440) is reproduced with a reasonable accuracy. The obtained structure of one-meson exchange interaction confirms a prediction of the large Nc limit approach stating that the mass splitting between various baryon states receives contributions from operators which simultaneously couple spin, isospin and orbital momentum. It is shown that one-meson exchange interaction generates a splitting between negative parity N*(1/2−) and N*(3/2−) states, and also between Δ*(3/2−) and Δ*(1/2−) states in contrast to the non-relativistic Goldstone boson exchange based quark models.

Core-level electronic spectra in ADC(2) approximation for polarization propagator: Carbon monoxide and nitrogen molecules

An effective ab initio approach to core-level electronic spectral studies is discussed. The approach uses polarization propagator theory in a second-order algebraic diagram construction ADC(2) approximation for calculating the characteristics of electron transitions; it also uses the linear vibronic model LVM for investigating the vibrational structure of transitions. The core excitation specialization of ADC(2) is achieved by introducing the core valence separation (CVS) approximation. K-excitation spectra of CO and N2 molecules are calculated to examine the potential of the approach. The calculated spectra and the available experimental data are analyzed to characterize the method. A number of additional facts of methodological and practical value are found, and new transitions are predicted. It is concluded that ADC(2)/CVS/LVM is a promising approach to problem solving in core level spectroscopy, which requires qualitatively reliable theoretical estimations.

Estimate of the second order electron self energy corrections in highly charged heavy ions

An estimate for the last unknown gauge-invariant set of QED corrections of order 2 ,t he second-order self-energy correction, is presented utilizing the so-called sign approximation. This is able to reduce the present uncertainties in Lamb-shift predictions considerably. Highly charged ions provide an ideal scenario to demonstrate the validity of QED in strong fields, e.g., by measurements of the Lamb shift at utmost precision. In this respect the recent experimental progress made in measurements in hydrogen-like uranium [1] indicates that calculations of higher-order radiative corrections become relevant. The set of second-order QED diagrams includes all various combinations of the first-order self-energy (SE) and vacuum polarization (VP) effects. The present status of theoretical predictions for the Lamb shift in different one-electron ions is presented in [2]. However, the calculations of the most difficult set of diagrams (see figure 1), the second-order self-energy correction (SESE) are yet uncomplete [3‐ 6], which remains as source for major theoretical uncertainties in present Lamb-shift predictions. To close this gap is a challenge for theory. As a step towards this, we present an estimate for the two-photon self-energy contribution. The graph SESE (a) consists of the irreducible and the reducible contributions. Utilizing Feynman gauge the first part can be renormalized and calculated separately since it does not contain infrared divergencies. The irreducible part has been calculated in [3] for large nuclear charge numbers Z and in [4,6] for arbitrary values of Z .F or high-Z systems all results obtained coincide, while for low- and intermediate-Z values a discrepancy between [4] and [6] has been found. In Feynman gauge, the remaining reducible part of the SESE (a) graph and the two diagrams SESE (b) and (c) have to be calculated simultaneously in order to cancel infrared and ultraviolet divergences arising from different diagrams [7]. The general renormalization scheme for all two-photon self-energy diagrams has

Surface interpolation based on new local coordinates

Sibson found a “local coordinate property” associated with the Voronoi diagram, and applied it to the surface interpolation from given heights at arbitrarily located sites. This article presents another local coordinate property that is also based on the Voronoi diagram. Our new property is simpler in that the coordinates can be computed from the Voronoi diagram directly while Sibson’s coordinates require the second-order Voronoi diagram. Moreover, our formula is flexible in the sense that it is valid even if the structure of the Voronoi diagram is incorrectly recognized. On the basis of this formula, a new interpolation scheme is constructed.

RELATIVISTIC MANY-BODY CALCULATIONS OF ENERGIES OFn= 3 STATES FOR THE BORON ISOELECTRONIC SEQUENCE,Z= 6–30

Energies of the 2s2(1S)3l, 2s2p(1P)3l, 2s2p(3P)3l, 2p2(3P)3l, 2p2(1D)3l, and 2p2(1S)3l states, withl = s,p, andd, for B-like ions with nuclear chargesZin the range 6–30 are determined to second order in relativistic many-body perturbation theory starting from a frozen-core Dirac–Fock basis. These perturbation theory calculations include the first- and second-order diagram contributions from the Coulomb and Breit operators for diagonal and nondiagonal matrix elements. Corrections for the frequency dependence of the Breit interaction are taken into account in lowest order. The Lamb-shift correction to the energies is also included in lowest order. Contributions of the three-electron interaction are compared with contributions of the one- and two-electron interactions. Comparisons are made with other theoretical calculations and with experimental data. We obtain excellent agreement for term splitting, even for low-Zions.

Lifetime of bound states of negative pions and neutrons

We estimate the lifetime of systems consisting of negative pions and neutrons (pineuts). We consider the direct decay of the pion as well as second-order diagrams involving the intermediate decay processes $n\ensuremath{\rightarrow}{p+e}^{\ensuremath{-}}+\overline{\ensuremath{\nu}}$ and ${\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{+e}^{\ensuremath{-}}+\overline{\ensuremath{\nu}}$ by assuming a simple single-particle model. We found that the dominant process is that with an intermediate neutron decay which leads to a lifetime of around ${10}^{\ensuremath{-}8}$ sec.