Combinatorial Hopf algebras from renormalization

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.Comment: 16 page

On the trees of quantum fields

The solution of some equations involving functional derivatives is given as a series indexed by planar binary trees. The terms of the series are given by an explicit recursive formula. Some algebraic properties of these series are investigated. Several examples are treated in the case of quantum electrodynamics: the complete fermion and photon propagators, the two-body Green function, and the one-body Green function in the presence of an external source, the complete vacuum polarization, electron self-energy and irreducible vertex.Comment: 14 pages. LaTeX file with FeynMF commands. Corrected version (17 June 99

A differential identity for Green functions

If P is a differential operator with constant coefficients, an identity is derived to calculate the action of exp(P) on the product of two functions. In many-body theory, P describes the interaction Hamiltonian and the identity yields a hierarchy of Green functions. The identity is first derived for scalar fields and the standard hierarchy is recovered. Then the case of fermions is considered and the identity is used to calculate the generating function for the Green functions of an electron system in a time-dependent external potential.Comment: 14 page

The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.Comment: 21 pages, 15 figure

Site symmetry and crystal symmetry: a spherical tensor analysis

The relation between the properties of a specific crystallographic site and the properties of the full crystal is discussed by using spherical tensors. The concept of spherical tensors is introduced and the way it transforms under the symmetry operations of the site and from site to site is described in detail. The law of spherical tensor coupling is given and illustrated with the example of the electric dipole and quadrupole transitions in x-ray absorption spectroscopy. The main application of the formalism is the reduction of computation time in the calculation of the properties of crystals by band structure methods. The general approach is illustrated by the examples of substitutional chromium in spinel and substitutional vanadium in garnet.Comment: 27 pages, 3 figure

Measurement of Magnetic Moment at the Atomic Scale in a High TC Molecular Based Magnet

The molecular-based magnet Cs^(I) [Ni^(II) Cr^(III) (CN)6]-2H2O is a ferromagnetic with a Curie temperature TC ) 90 K. Its structure consists of face-centered cubic lattice of Ni^(II) ions connected by Cr(CN)6 entities. We have recorded X-ray magnetic circular dichroism (XMCD) at nickel L2,3 edges. It clearly evidences that nickel(II) is in a high-spin configuration and ferromagnetically coupled to the surrounding Cr^(III) . Through ligand field multiplet calculations, we have determined the total magnetic moment carried by Ni^(II) . Special attention has been paid to the magnetic anisotropy that complicates the calculation of the cross section for a powder. By using sum rules derived for XMCD, it has been possible to extract the orbital and spin contributions to the total magnetic moment. A somewhat too small magnetic moment is found on nickel. A complete calculation taking into account the multiplet coupling effect and the covalent hybridization allowed to determine the precise ground state of nickel and showed that hybridization cannot be responsible for the experimental low nickel magnetic moment. The origin of this effect is discussed

Assessing temperature effects on multipole contributions and angular dependence in core-level spectroscopies

This study aims at assessing the thermal nuclei motion effects on the multipole transition channels involved in two core-level spectroscopies, x-ray absorption spectroscopy (XAS) and x-ray Raman scattering (XRS). Temperature effects on the 1s -> s monopole, 1s -> p dipole, and 1s -> d quadrupole transitions are investigated using two reference systems for which we present original experimental data: alpha-Al2O3 at the Al K edge probed by XRS at room temperature and rutile TiO2 at the Ti K pre-edge probed by XAS at temperatures ranging from 6 to 700 K. Through the latter, this work enlightens the part of the pre-edge peak enhancement due to temperature in the K pre-edge region of 3d transition metal, which is known to be routinely used to determine the concentration, valence or symmetry of the probed element in a given sample. Nuclear thermal fluctuations are taken into account using a method based on density functional theory that consists in averaging spectra over atomic configurations, generated within the harmonic approximation and obeying quantum statistics at finite temperature. Since only a finite number of such configurations are used, the numerically averaged spectra generally lose the symmetry of the equilibrium crystal positions. In this paper, we demonstrate that the physical average has to be symmetric and propose a method to restore the physical angular dependence of the spectra. The approach is successfully applied to investigate the angular dependent XAS spectra in rutile as a function of temperature. The two systems under study allow to draw general conclusions regarding the effect of nuclear quantum fluctuations on the different transition channels available to both core-level spectroscopies.Peer reviewe

Quantum field theory and Hopf algebra cohomology

We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.Comment: 39 pages, no figures, LaTeX + AMS macros; title changed, minor corrections, references update

Experimental evidence of thermal fluctuations on the X-ray absorption near-edge structure at the aluminum K-edge

After a review of temperature-dependent experimental x-ray absorption near-edge structure (XANES) and related theoretical developments, we present the Al K-edge XANES spectra of corundum and beryl for temperature ranging from 300K to 930K. These experimental results provide a first evidence of the role of thermal fluctuation in XANES at the Al K-edge especially in the pre-edge region. The study is carried out by polarized XANES measurements of single crystals. For any orientation of the sample with respect to the x-ray beam, the pre-edge peak grows and shifts to lower energy with temperature. In addition temperature induces modifications in the position and intensities of the main XANES features. First-principles DFT calculations are performed for both compounds. They show that the pre-edge peak originates from forbidden 1s to 3s transitions induced by vibrations. Three existing theoretical models are used to take vibrations into account in the absorption cross section calculations: i) an average of the XANES spectra over the thermal displacements of the absorbing atom around its equilibrium position, ii) a method based on the crude Born-Oppenheimer approximation where only the initial state is averaged over thermal displacements, iii) a convolution of the spectra obtained for the atoms at the equilibrium positions with an approximate phonon spectral function. The theoretical spectra so obtained permit to qualitatively understand the origin of the spectral modifications induced by temperature. However the correct treatment of thermal fluctuation in XANES spectroscopy requires more sophisticated theoretical tools