Quantum Interference, Graphs, Walks, and Polynomials

Dyadic Green's functions for inhomogeneous parallel-plate waveguides are considered.The usual residue series form of the Green's function is examined in the case of modal degeneracies, where secondorder poles are encountered.The corresponding second-order residue contributions are properly interpreted as representing "associated functions" of the structure by constructing a new dyadic root function representation of the Hertzian potential Green's dyadic.The dyadic root functions include both eigenfunctions (corresponding to first-order residues) and associated functions, analogous to the idea of Jordan chains in finite-dimensional spaces.Numerical results are presented for the case of a two-layer parallel-plate waveguide.

Few-body problems involving Coulomb or gravitational interactions between pairs of particles, whether in classical or quantum physics, are generally handled through a standard multipole expansion of the two-body potentials. We discuss an alternative based on a compact, cylindrical Green's function expansion that should have wide applicability throughout physics. Two-electron "direct" and "exchange" integrals in many-electron quantum systems are evaluated to illustrate the procedure which is more compact than the standard one using Wigner coefficients and Slater integrals.

In the paper, we study the two-loop contribution to the effective action of the four-dimensional quantum Yang-Mills theory. We derive a new formula for the contribution in terms of three functions, formed from the Green's function expansion near the diagonal. This result can be applied to different types of regularization. Therefore, we test it by using the dimensional regularization and cutoff ones and show the consistence with the results, obtained in other works.

The heterogeneity of the subsurface permeability is considered as the most influential factor in determining groundwater flow and the transport of toxic contaminants. Numerical simulators cannot handle the large grids required to represent the small-scale variability of permeability, and thus explicit estimates of the large-scale behavior in terms of coarse-grained parameters are often required. Perturbation formulations of the effective permeability are based on simplifying assumptions that are valid only for certain probability distributions and weak heterogeneity. A generalized perturbation ansatz that involves higher orders has been proposed (Gelhar and Axness, 1983), but to our knowledge its validity has not been rigorously proved before in three dimensions. In this work we propose a general upscaling formulation valid for strong heterogeneity, general permeability distributions, and media with impermeable zones. We show that the effective permeability is determined by the self-energy series of the permeability fluctuations at zero frequency. Using the diagrammatic representation, we obtain a Dyson equation that involves only irreducible diagrams of the proper self-energy series. We develop a renormalization group (RG) analysis for isotropic lognormal media that proves the generalized perturbation ansatz to all orders. We show that the RG result accurately estimates laboratory permeability measurements in limestone (strong heterogeneity) and sandstone (weak heterogeneity). We also propose an explicit RG estimate for the preasymptotic effective permeability. We compare our results with an approach based on a leading order Green's function expansion (Paleologos et al., 1996), which, however, requires intensive numerical computations. Finally, we investigate the relation between the RG expression and the algebraic means used in numerical upscaling.

A novel image-series expansion scheme for quasi-static Green's function in n+1 layered media is obtained by expanding the frequency-dependent Hertz potential in finite expansions and remainder terms. The expansions utilize a unique recursive representation for Green's function, which is a generic characteristic of the stratification, and are explicitly constructed for n/spl les/3. While results for 0/spl les/n/spl les/2 are given for reference only, the expansion scheme for a double-slab configuration, n=3, is quite general and outlines the procedure for n>3, without any increase in the complexity. The expansion-remainder terms can be made negligibly small for sufficiently large summation indices in the quasi-static limit, leading to rigorous image-series expansion. The image-series convergence is accelerated by including a collective image term, representing a closed-form asymptotic evaluation of the series-remainder integral. Thus, the proposed computational procedure can be used as a simple tool for producing analytical data for testing numerical subroutines applied to direct problems such as electrical simulation of muscles in the biomedical field and inverse problems, such as electromagnetic imaging.

Shockley surface states (SS) have attracted much attention due to their role in various physical phenomena occurring at surfaces. It is also clear from experiments that they can play an important role in electron transport. However, accurate incorporation of surface states in $\textit{ab initio}$ quantum transport simulations remains still an unresolved problem. Here we go beyond the state-of-the-art non-equilibrium Green's function formalism through the evaluation of the self-energy in real-space, enabling electron transport without using artificial periodic in-plane conditions. We demonstrate the method on three representative examples based on Au(111): a clean surface, a metallic nanocontact, and a single-molecule junction. We show that SS can contribute more than 30\% of the electron transport near the Fermi energy. A significant and robust transmission drop is observed at the SS band edge due to quantum interference in both metallic and molecular junctions, in good agreement with experimental measurements. The origin of this interference phenomenon is attributed to the coupling between bulk and SS transport channels and it is reproduced and understood by tight-binding model. Furthermore, our method predicts much better quantized conductance for metallic nanocontacts.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

We study the energy structure and dynamics of a V-type three-level emitter embedded in a one-dimensional waveguide with model dispersion. Due to the presence of two different couplings between the upper levels and the waveguide modes, we find that the bound states in this system are quite different from the case of a two-level emitter. The energy structure is dependent on the relative position of the emitter's transition frequencies from the band with finite width. We obtain the phase diagrams through numerical analysis. The dynamics of spontaneous emission of the system is calculated exactly in the general case by means of the Green's function approach. The emitted photon is characterized by two components: a localized part and a traveling part, and the average frequency of the traveling part is different from that for resonant scattering. Through the quantum interference between the emitter's two transitions, it is found that the total excitation number of the localized photon and emitter can be greatly enhanced in some case in the spontaneous-emission process.

The Feynman diagrams of the Green's function expansion of fermions interacting with a nonrelativistic 2‐body interaction are displayed in first, second, and third perturbative order of the interaction as 2, 10, and 74 diagrams, respectively. A name convention for the diagrams is proposed and then used to tabulate the diagrams of fourth to seventh order. The Hartree–Fock approximation summons up 2, 8, 40, and 224 of them in first through fourth order. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

By employing the Green's-function expansion method, the optical spectrum rearrangement in the quasi-one-dimensional (1D) impurity vibrational systems has been investigated. Based on the validity criteria for the conventional coherent potential approximation (CPA), the dispersion of phononlike modes and main parameters for the rearranged spectra are determined. The most interesting rearrangement occurs when the bandwidth exceeds the limiting frequency of the optical branch. In this case, the spectrum rearrangement exhibits features characteristic of both the acoustic branch and high-frequency optical branch. Formations of the impurity optical branch and a well-defined limiting frequency are described.

A semi-analytical approach for rapid calculation of the vector potential normal modes and dyadic Green's function inside the cavity of irregular shape over a broad range of frequency is presented. The method is based on the extraction of the Green's function at an imaginary wave number from itself to obtain a rapidly convergent hybrid spatial-spectral expansion of the Green's function. The extracted term encompasses the singularity of the Green's function and is computed using surface integral equation. The method is applied to a V-grooved cavity to obtain the vector potential Green's function inside the cavity.

The dyadic Green's functions of the cylindrical chirowaveguide are derived by modal expansion. Bohren's decomposition of the electromagnetic field is used to obtain the vector wave functions. The magnetic dyadic Green's function that is purely solenoidal is first derived and the electric dyadic Green's function is then obtained by manipulating the magnetic dyadic Green's function. The singular term in the expression of the electric dyadic Green's function is reinstated through the manipulation procedure.

Quantum interference (QI) is well recognised as a significant contributing factor to the magnitude of molecular conductance values in both single-molecule and large area junctions. Numerous structure-property relationship studies have shown that para-connected oligo(phenyleneethynylene) (OPE) based molecular wires exemplify the impact of constructive quantum interference (CQI), whilst destructive quantum interference (DQI) effects are responsible for the orders of magnitude lower conductance of analogous meta-contacted OPE derivatives, despite the somewhat shorter effective tunnelling distance. Since molecular conductance is related to the value of the transmission function, evaluated at the electrode Fermi energy, T(EF), which in turn is influenced by the presence and relative energy of (anti)resonances, it follows that the relative single-molecule conductance of para- and meta-contacted OPE-type molecules is tuned both by the anchor group and the nature of the electrode materials used in the construction of molecular junctions (gold|molecule|gold vs. gold|molecule|graphene). It is shown here that whilst amine-contacted junctions show little influence of the electrode material on molecular conductance due to the similar electrode-molecule coupling through this anchor group to both types of electrodes, the weaker coupling between thiomethyl and ethynyl anchors and the graphene substrate electrode results in a relative enhancement of the DQI effect. This work highlights an additional parameter space to explore QI effects and establishes a new working model based on the electrode materials and anchor groups in modulating QI effects beyond the chemical structure of the molecular backbone.

Aided by density functional theory with nonequilibrium Green's functions simulations, the thermoelectric (TE) properties of nanoporous carbon nanotube junctions formed by covalently binding carbon nanotubes via benzene bridges with either meta‐ or para‐connections are investigated. The results show that the TE performances of meta‐connected nanoporous carbon nanotube junctions are significantly improved due to the influence of destructive quantum interference (DQI) effect. Moreover, the TE properties of meta‐connected carbon nanotube junctions can be further improved by substitution of nitrogen atom and gate voltage. The reason is that the quantum interference patterns can be transformed from DQI to constructive quantum interference (or Fano resonance). The theoretical analysis shows that the TE figure of merit (ZT) for meta‐connected carbon nanotube junctions at room temperature is close to 0.5 near the Fermi level, ≈20 times higher than that of the perfect carbon nanotube junction. These results reveal that quantum interference effect can indeed be used to enhance the TE performance of molecular junctions.

Ab initio electron propagator methods are developed to study electronic properties of molecular systems with strong electron-electron and electron-phonon interactions. For the calculation of electron Green's functions we apply a canonical small polaron transformation that intrinsically contains strong electron-phonon effects. In the transformed Hamiltonian, the energy levels for the noninteracting particles are shifted down by the relaxation (solvation) energies. The Coulomb integrals are also renormalized by the electron-phonon interaction. For certain values of the electron-phonon coupling constants, the renormalized Coulomb integrals can be negative which implies the attraction between two electrons. Within the small polaron transformation we develop a diagrammatic technique for the calculation of electron Green's function in which the electron-phonon interaction is already included into the multiple phonon correlation functions. Since the decoupling of the phonon correlation functions is impossible, and therefore, a Wick's theorem for such correlation functions is invalid, there is no Dyson equation for the electron Green's function. To find the electron Green's function, we use different approximations. One of them is a link-cluster approximation that includes diagonal transitions for the renormalized zeroth Green's function. In the linked-cluster approach the Dyson equation is derived in the most general case, where the self-energy operator is an arbitrary functional (not only in the Hartree-Fock approximation). It is shown that even a Hartree-Fock electron (hole) is not a particle any longer. It is a quasiparticle with a finite lifetime that depends on energy of particle and hole states in different ways. As a consequence of this, a standard description of a Hartree-Fock approximation in terms of wave functions becomes inappropriate in this problem. To challenge the linked-cluster approximation we develop a different approach: a sequential propagation approximation where scattering events occur only for sequential transitions. A self-consistent Hartree-Fock equation for a four-index Green's function matrix is derived. In conclusion, the proposed schemes can be considered for future method developments for quantum chemical calculations for large molecules with strong nonadiabatic effects, e-e correlated electron transfer reactions, and electron transport in molecular transport junctions.