Sum Of Diagrams Research Articles

Features of a theoretical description of ionization potentials of the LiF molecule and electron affinity of the F atom

We have performed calculations of the ionization potentials of the LiF molecule and the electron affinity of the F atom (as the ionization potential of F−) by the Green's function method in the approximation adapted for calculation of the ionization potentials for ionization from upper MO's and the ADC(3) approximation. We have shown that quantitative agreement with experimental values can be obtained only as a result of infinite partial summation of diagrams in the ADC(3) scheme.

The fully frustrated Ising model in infinite dimensions

The authors solve, subject to the validity of some reasonable assumptions, the 'fully frustrated' Ising model in the limit of infinite dimensions using an extension of the TAP theory for spin glasses. In contrast to the TAP theory of the infinite-range spin glass, an infinite summation of diagrams is required to recover the Gibbs free energy for this model. The model undergoes a first-order transition. The method used to solve the model should have many applications to other physical problems.

Comment on "Ring diagram nuclear matter calculations using Bonn and V14 potentials"

The average binding energy per nucleon in nuclear matter had recently been calculated in the framework of a model space approach, including the sum of the particle-particle and hole-hole ring diagrams to all orders. It is pointed out that, in the case of the ${V}_{14}$ interaction, the calculated saturation properties are at variance with an upper bound evaluated from a variational method; they are also quite different from those derived from the standard hole-line expansion. It is suggested that these differences might be due to bubble contributions that have, in part, been omitted in the model space approach. It is also proposed to use the summation of particle-particle and hole-hole ring diagrams to test the convergence of the standard hole-line expansion.

Particle-particle propagator in the algebraic diagrammatic construction scheme at third order.

The algebraic diagrammatic construction scheme at nth order includes all Feynman diagrams up to nth order and a partial summation of diagrams arising from the latter to infinite order. In the present work this scheme is carried out for the first time at third order for the particle-particle propagator. Because of the many diagrams involved in third order the approach requires a very careful analysis of terms which describe the corresponding perturbation expansion of the propagator. The relevant technical details of the analysis are presented.

Multiple scattering by discrete random media: a summation of planar diagrams

The multiple scattering of a scalar wave by a discrete random medium is considered. The scatterers are assumed to be isotropic, incorrelated, and distributed uniformly. The Feynman-diagram technique is used to formulate the problem, and the Dyson equation is derived with the occupation-number formalism. Among the terms of the mass operator, a topological class of 'planar diagrams' is defined, excluding all the interactions involving the same particle three times or more. It is possible to achieve analytically the summation of such diagrams. The corresponding mean field is computed and compared with other approximations such as those of Born and Twersky. The result does not reduce to Twersky's even in the case of small concentrations of scatterers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Toward a simple density functional theory of nonuniform solids

With analogy to the “highly accurate” summation of cluster diagrams for hard sphere fluids à la Carnahan-Starling, a simple real space free-energy density functional for arbitrary potential systems is proposed, based on a generalization of the second virial coefficient to inhomogeneous systems, which when applied to ordered and amorphous solid hard-sphere systems yields pressures in remarkable agreement with experiment. Possibilities for corrections and extensions toward a simple density functional theory of nonuniform solids are noted.

Cut-off parameters of the bosonisation technique in one dimension

The Jordan boson representation and point-splitting regularisation are applied to the one-dimensional two-fermion model (TFM) with momentum-transfer cut-off and zero-mode terms included. It is shown that the correlation functions of the Tomonago-Luttinger model obtained in this way coincide with those given by diagram summation. The energy gaps entering the well known Luther-Emery-Peschel back-scattering and Umklapp scattering solutions acquire a physical dependence on the momentum-transfer cut-off. First-order perturbation calculations of the charge-density function give different results for the spin-parallel 'back-scattering' contribution to the bosonised TFM and corresponding Fermi gas model, thus indicating that these models are not equivalent.

Summation of ring diagrams of nuclear thermodynamic potential

Using an imaginary-time linked diagram expansion of the thermodynamic potential Ω, we derive methods by which the particle-particle (hole-hole) and the particle-hole ring diagrams of Ω can each be summed up to all orders in a rather convenient way. A model space P and a corresponding finite temperature reaction matrix K are introduced in order to take care of the strong short-range correlations between nucleons. Methods of the finite temperature Green functions are employed. The contribution from the particle-particle (hole-hole) ring diagrams to Ω is given in terms of simple integrals of the form ∝ 0 1d λ tr p { M( λ) K} where λ is a strength parameter, and M a thermal transition matrix which is calculated from a finite-temperature RPA-type secular equation. Solutions of this equation are derived, and their possible connections with phase transitions are discussed. The particle-hole ring diagrams are treated in a similar way. Applications of the present approach and prospects for formulating a Brueckner-type theory of nuclear matter at finite temperature are discussed.

The relativistic two-nucleon problem in nuclear matter

A new covariant formalism is developed for studying two-baryon correlations in infinite nuclear matter. The formalism is based on renormalizable, relativistic quantum field theories of mesons and baryons (quantum hadrodynamics) and involves a self-consistent summation of the relativistic ladder diagrams using a quasipotential approximation. Self-consistency modifies both the singleparticle spectrum and wave functions and introduces important density dependence into matrix elements of the one-boson-exchange (ladder) interaction. Results are obtained for the Walecka model, which describes the nucleon-nucleon interaction through neutral scalar and vector meson exchange. The mean-field theory (MFT) values of the large scalar and vector components of the baryon self-energy (“single-particle potential”) are modified only slightly by correlations, yielding a baryon effective mass M ∗ that decreases rapidly with increasing density. Moreover, correlations have a small effect on the high-density nuclear equation of state, which is quite stiff due to the small value of M ∗ and is given accurately by the MFT. In contrast, due to cancellations near equilibrium density, correlations produce significant corrections to the MFT binding energy. Calculated saturation properties are sensitive to the choice of quasipotential and the introduction of meson-baryon form factors. Finally, modifications from the shifted baryon mass in negative-energy states are included in a mean-field approximation. These corrections are significant in calculations of the binding energy.

Infinite order summation of particle-particle ring diagrams in a model-space approach for nuclear matter

A ring diagram model-space nuclear matter theory is formulated and applied to the calculation of the binding energy per nucleon ( BE A ) , saturation Fermi momentum ( k F) and incompressibility coefficient ( K) of symmetric nuclear matter, using the Paris and Reid nucleon-nucleon potentials. A model space is introduced where all nucleons are restricted to have momentum k ⩽ k M, typical values of k M being ∼3.2 fm −. Using a model-space Hartree-Fock approach, self-consistent single particle spectra are derived for holes ( k ⩽ k F) and particles with momentum k F < k ⩽ k M. For particles with k > k M, we use a free particle spectrum. Within the model space we sum up the particle-particle ring diagrams (both forward- and backward-going) to all orders. A rather simple expression for the energy shift ΔE 0 pp is obtained, namely ΔE 0 pp is expressed as integrals involving the trace of Y( λ) Y +( λ) G M where G M is the model-space reaction matrix, the Y's are transition amplitudes obtained from solving RPA-type secular equations and λ is a strength parameter to be integrated from 0 to 1. We have used angle-average approximations in our calculations, and in this way different partial wave channels are decoupled. For the 3S 1- 3D 1 channel, the effect of the ring diagrams is found to be particularly important. The inclusion of the ring diagrams has largely increased the role of the tensor force in determining the nuclear matter saturation properties, and consequently we obtain saturation densities which are significantly lower than those given by most other calculations. For the Paris potential, our results for BE A , k F and K are respectively 17.38 MeV, 1.42 fm −1 and 96.3 MeV. For the Reid potential, the corresponding results are 15.15 MeV, 1.30 fm −1 and 110.7 MeV. Our calculated values for the binding energy per nucleon and saturation density are both in rather satisfactory agreement with the corresponding empirical values.

Connected moments of the Hamiltonian in SU(3) lattice gauge theory.

We calculate expectation values of powers of the Hamiltonian of the SU(3) lattice gauge theory in the strong-coupling ground state. The moments are written as a sum of diagrams. We describe the algorithm used to evaluate the diagrams numerically.

Magnetic Field Induced Localization in Three Dimensional Metallic Systems

Applying the self-consistent treatment of the Anderson localization problem to three dimensional systems under strong magnetic fields, we calculate the anisotropic static diffusion coefficients in the case when the Fermi level is within the lowest Landau subband. Two coupled self-consistent equations are derived through partial summation of diagrams for two diffusion coefficients D // and D ⊥ which describe the electronic diffusion along and perpendicular to the field, respectively. They decrease with increasing magnetic field and vanish linearly at a critical field which is determined by ε F 0 and τ 0 , the zero field value of the Fermi energy and the elastic relaxation time, the ratio D ⊥ / D // remaining finite and non-vanishing. The cyclotron frequency ω crt corresponding to the critical field is found to satisfy (ε F 0 /ω crt )(ε F 0 τ 0 ) 1/4 =0.509.

Electric microfield distributions in plasmas of arbitrary degeneracy and density.

A new method of calculating electric microfields in plasmas, based on density-functional theory (DFT), is presented. Strong-coupling effects, quantum effects, internal structure of the test particle, etc., are conveniently and rigorously treated. DFT is used to calculate the ion-electron and ion-ion pair distribution functions ${g}_{\mathrm{ie}}$ and ${g}_{\mathrm{ii}}$. These are then used to calculate the electron-screened electric field and the first- and second-order terms in the Baranger-Mozer (BM) formulation of the microfield probability function. When low-order BM fails for strongly coupled systems, a weighted summation of chainlike diagrams may be used for evaluating the all-order sum. We present low-frequency microfield calculations for neutral and charged test particles without structure, a neutral particle with H-like structure, and an aluminum ion immersed in a hydrogen plasma. We also consider the case of neutral and charged test points in an aluminum plasma and compare our results with apex. Our microfields differ significantly from the usual ones, and depend specifically on the nature of the plasma, as well as the nature and structure of the test particle. For increasingly classical weakly coupled plasmas our results fall into agreement with the existing microfields which depend essentially on the ratio (density${)}^{\mathrm{\ensuremath{-}}1/3}$/(Debye length).

Summation of particle-particle and particle-hole ring diagrams in binding energy calculations and Lipkin models

In this paper we study methods for calculating the particle-particle (pp) and particle-hole (ph) ring diagrams in the linked diagram expansion of the ground-state energy shift Δ E 0 for many-body systems. Analytic expressions for summing up the pp ring diagrams to all orders are derived, given as an integral involving G pp( ω, λ) ν, where G pp is the pp Green function, υ the interaction hamiltonian (taken to be two-body) and λ a strength parameter to be integrated over from 0 to 1. Similar expressions for summing up the ph ring diagrams are also derived, their integral form involving G ph (ω, λ) ν, where G ph is the ph Green function and ν the particle-hole interaction. T energy variable ω is to be integrated over from −∞ to +∞. Two methods for carrying out the ω-integration are suggested. One is the multipole expansion method, and the other is the transition amplitude method where the Green functions G ppand G ph are calculated using RPA equations from which the physical transition amplitudes can be obtained. Our methods for summing up the pp and ph ring diagrams to all orders are applied to several Lipkin-model many-body problems. The effect of using Hartree-Fock self-consistent single-particle wave functions on the contribution of high-order ring diagrams to Δ E 0 is studied and found to be very important. Possible applications of our methods to actual nuclear many-body problems are discussed.

Theory of inhomogeneous quantum systems. I. Static properties of Bose fluids.

We develop a variational theory for the determination of the ground-state properties of an inhomogeneous Bose fluid at zero temperature. Euler-Lagrange equations are derived for the one- and two-body correlations, and systematic approximation methods are physically motivated. It is shown that the optimized variational method provides a self-consistent approximate summation of ladder and ring diagrams. Numerical applications are presented for the density profile, distribution functions, energy, and chemical potential of films of $^{4}\mathrm{He}$ atoms. Both the bulk and surface energies are calculated and found to be in fair agreement with experiments.

Two-nucleon correlations in a relativistic theory of nuclear matter

Relativistic two-nucleon correlations are studied using a self-consistent summation of ladder diagrams in nuclear matter. Correlations have a small effect on the nucleon self-energy but may produce significant changes in the binding energy. Correlation corrections to the mean-field equation of state are small at high density.

Renormalization of the self-consistent summation of exchange diagrams in a relativistic quantum field theory

The self-consistent summation of exchange diagrams is renormalized in a relativistic fermionscalar field theory using spectral function methods. This is a non-perturbative approach in which all uncrossed exchange diagrams are summed to all orders in the coupling constant. A detailed exposition of the renormalization technique is presented along with numerical results. Anomalous analytic structure (ghost poles and ghost branch cuts) that arises naturally from the approximation is briefly discussed. We discuss the generalization to a finite density of fermions that may have application in nucleon-meson field theories which are currently in vogue.

The approximate summation of planar diagrams in matrix ϕ 4-models

A method of approximate summation of planar diagrams in matrix ϕ 4-models is proposed. In the case of zero- and one-dimensional models the results are in good agreement with the known results of Brezin et al.

Coupled-cluster summation of the particle-particle ladder diagrams for the two-dimensional electron gas

The correlation energy and antiparallel spin pair correlation function of the two-dimensional electron gas are studied within the particle-particle ladder approximation of the coupled-cluster equations. Unlike the three-dimensional electron gas the correlation energy is shown to be convergent in the particle-particle ladder approximation for the two-dimensional system. The coupled-cluster particle-particle ladder equations are cast into the form of singular integral equations. Using a product integration technique the integral equations are solved numerically. The correlation energy and r=0 value of the antiparallel spin pair correlation function are calculated from the numerical coupled-cluster coefficients for a range of electron densities from rs=0 to 16. The antiparallel spin pair correlation function at r=0 is shown to be positive for all electron densities when a wavefunction accurate to first order in the coupled-cluster particle-particle ladder coefficients is used. The calculated correlation energy is compared with dielectric results of Jonson (1976) and ring results of Freeman (1977, 1978). Good agreement between the dielectric results and the particle-particle ladder results are found for intermediate to low electron densities. The particle-particle ladder coupled-cluster correlation energy is shown to be exact in the high-density limit, although the approach to the limit is not correct owing to the neglect of ring contributions. It is suggested that simultaneous inclusion of particle-particle ladder and ring diagrams will give correlation energies accurate over the entire range of calculated densities.

Weak absence of diffusion in two dimensions in the weak-disorder limit

Working within the single-electron picture of two-dimensional disordered systems in the crucial weak-disorder regime, the authors calculate the average two-particle Green function by summing the maximally crossed diagrams occurring in the approximation of Neal and Langer (1966) as developed by Abrahams and Ramakrishnan (1980) to show that Anderson's criterion for location is not satisfied, while there is only a weak absence of diffusion in the sense of Ishii (1973). Summation of ladder diagrams appearing in the Langer-Neal scheme gives an identical result in the small-disorder regime. They reach the same conclusion upon using the CPA to perform the required averaging. Recent mathematical results associate such a weak absence of diffusion with a singularly continuous spectrum.