arXiv subjects

Christian Brouder

Publications and source records attributed to Christian Brouder.

At least 37 records · Page 2Linked to original sources

arXiv2012

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.

cond-mat.str-el↗

arXiv2011

Nonlocal, noncommutative diagrammatics and the linked cluster Theorems

Recent developments in quantum chemistry, perturbative quantum field theory, statistical physics or stochastic differential equations require the introduction of new families of Feynman-type diagrams. These new families arise in various ways. In some generalizations of the classical diagrams, the notion of Feynman propagator is extended to generalized propagators connecting more than two vertices of the graphs. In some others (introduced in the present article), the diagrams, associated to noncommuting product of operators inherit from the noncommutativity of the products extra graphical properties. The purpose of the present article is to introduce a general way of dealing with such diagrams. We prove in particular a "universal" linked cluster theorem and introduce, in the process, a Feynman-type "diagrammatics" that allows to handle simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams arising from the study of interacting systems (such as the ones where the ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or electric field, by impurities...) or Wightman fields (that is, expectation values of products of interacting fields). Our diagrammatics seems to be the first attempt to encode in a unified algebraic framework such a wide variety of situations. In the process, we promote two ideas. First, Feynman-type diagrammatics belong mathematically to the theory of linear forms on combinatorial Hopf algebras. Second, linked cluster-type theorems rely ultimately on Möbius inversion on the partition lattice. The two theories should therefore be introduced and presented accordingly

math-ph↗

arXiv2011

One-particle irreducibility with initial correlations

In quantum field theory (QFT), the vacuum expectation value of a normal product of creation and annihilation operators is always zero. This simple property paves the way to the classical treatment of perturbative QFT. This is no longer the case in the presence of initial correlations, that is if the vacuum is replaced by a general state. As a consequence, the combinatorics of correlated systems such as the ones occurring in many-body physics is more complex than that of quantum field theory and the general theory has made very slow progress. Similar observations hold in statistical physics or quantum probability for the perturbation series arising from the study of non Gaussian measures. In this work, an analysis of the Hopf algebraic aspects of quantum field theory is used to derive the structure of Green functions in terms of connected and one-particle irreducible Greeen functions for perturbative QFT in the presence of initial correlations.

cond-mat.str-el↗

arXiv2010

Comment on "Delayed luminescence of biological systems in terms of coherent states" [Phys. Lett. A 293 (2002) 93]

Popp and Yan [F. A. Popp, Y. Yan, Phys. Lett. A 293 (2002) 93] proposed a model for delayed luminescence based on a single time-dependent coherent state. We show that the general solution of their model corresponds to a luminescence that is a linear function of time. Therefore, their model is not compatible with any measured delayed luminescence. Moreover, the functions that they use to describe the oscillatory behaviour of delayed luminescence are not solutions of the coupling equations to be solved.

q-bio.QM↗

arXiv2010

The Rayleigh-Schrödinger perturbation series of quasi-degenerate systems

We present the first representation of the general term of the Rayleigh-Schrödinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. The relation with several combinatorial objects used for special cases (degenerate or non-degenerate systems) is established.

quant-ph↗

arXiv2010

Quantum field theory meets Hopf algebra

This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S-matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and lead to a correspondence between Feynman diagrams and semi-standard Young tableaux. Reciprocally, these concepts are used as models to derive Hopf algebraic constructions such as a connected coregular action or a group structure on the linear maps from S(V) to V. In most cases, noncommutative analogues are derived.

hep-th↗

arXiv2010

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 Faà 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.

math.QA↗

arXiv2010

Tree expansion in time-dependent perturbation theory

The computational complexity of time-dependent perturbation theory is well-known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams...). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and non-perturbatively the convergence of Morita's generalization of the Gell-Mann and Low wavefunction. We show that summing all the terms associated to the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the time-independent equation for the wave operator and we give an explicit non-recursive expression for the term corresponding to an arbitrary tree.

cond-mat.str-el↗

arXiv2009

Angular dependence of core hole screening in LiCoO2: A DFT+U calculation of the oxygen and cobalt K-edge x-ray absorption spectra

Angular dependent core-hole screening effects have been found in the cobalt K-edge x-ray absorption spectrum of LiCoO2, using high-resolution data and parameter-free GGA+U calculations. The Co 1s core-hole on the absorber causes strong local attraction. The core-hole screening on the nearest neighbours cobalt induces a 2 eV shift in the density of states with respect to the on-site 1s-3d transitions, as detected in the Co K pre-edge spectrum. Our DFT+U calculations reveal that the off-site screening is different in the out-of-plane direction, where a 3 eV shift is visible in both calculations and experiment. The detailed analysis of the inclusion of the core-hole potential and the Hubbard parameter U shows that the core-hole is essential for the off-site screening, while U improves the description of the angular dependent screening effects. In the case of oxygen K-edge, both the core-hole potential and the Hubbard parameter improve the relative positions of the spectral features.

cond-mat.mtrl-sci↗

arXiv2009

Temperature dependence of the pre-edge structure in the Ti K-edge x-ray absorption spectrum of rutile

The temperature dependence of the pre-edge features in x-ray absorption spectroscopy is reviewed. Then, the temperature dependence of the pre-edge structure at the K-edge of titanium in rutile TiO2 is measured at low and room temperature. The first two peaks grow with temperature. The fact that these two peaks also correspond to electric quadrupole transitions is explained by a recently proposed theory.

cond-mat.mtrl-sci↗

arXiv2009

Gell-Mann and Low formula for degenerate unperturbed states

The Gell-Mann and Low switching allows to transform eigenstates of an unperturbed Hamiltonian $H_0$ into eigenstates of the modified Hamiltonian $H_0 + V$. This switching can be performed when the initial eigenstate is not degenerate, under some gap conditions with the remainder of the spectrum. We show here how to extend this approach to the case when the ground state of the unperturbed Hamiltonian is degenerate. More precisely, we prove that the switching procedure can still be performed when the initial states are eigenstates of the finite rank self-adjoint operator $\cP_0 V \cP_0$, where $\cP_0$ is the projection onto a degenerate eigenspace of $H_0$.

math-ph↗

arXiv2009

Effect of vibrations on the pre-edge features of x-ray absorption spectra

The influence of atomic vibrations on x-ray absorption near edge structure (XANES) is calculated by assuming that vibrational energies are small with respect to the instrumental resolution. The resulting expression shows that, at the K-edge, vibrations enable electric dipole transitions to 3s and 3d final states. The theory is applied to the K-edge of Al in Al2O3 corundum and of Ti in TiO2 rutile and compared with experiment. At the Al K-edge, sizeable transitions towards 3s final states are obtained, leading to a clear improvement of the agreement with experimental spectra. At the Ti K-edge, electric dipole transitions towards 3d final states explain the temperature dependence of the pre-edge features.

cond-mat.mtrl-sci↗

arXiv2009

The many-body Green function of degenerate systems

A rigorous non perturbative adiabatic approximation of the evolution operator in the many-body physics of degenerate systems is derived. This approximation is used to solve the long-standing problem of the choice of the initial states of H0 leading to eigenstates of H0+V for degenerate systems. These initial states are eigenstates of P0 V P0, where P0 is the projection onto a degenerate eigenspace of H0. This result is used to give the proper definition of the Green function, the statistical Green function and the non-equilibrium Green function of degenerate systems. The convergence of these Green functions is established.

cond-mat.str-el↗

arXiv2009

Hyperoctahedral Chen calculus for effective Hamiltonians

The algebraic structure of iterated integrals has been encoded by Chen. Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and Schützenberger. It is mostly incorporated in the modern theory of free Lie algebras. Here, we tackle the problem of unraveling the algebraic structure of computations of effective Hamiltonians. This is an important subject in view of applications to chemistry, solid state physics, quantum field theory or engineering. We show, among others, that the correct framework for these computations is provided by the hyperoctahedral group algebras. We define several structures on these algebras and give various applications. For example, we show that the adiabatic evolution operator (in the time-dependent interaction representation of an effective Hamiltonian) can be written naturally as a Picard-type series and has a natural exponential expansion.

math-ph↗

arXiv2008

Adiabatic approximation, Gell-Mann and Low theorem and degeneracies: A pedagogical example

We study a simple system described by a 2x2 Hamiltonian and the evolution of the quantum states under the influence of a perturbation. More precisely, when the initial Hamiltonian is not degenerate,we check analytically the validity of the adiabatic approximation and verify that, even if the evolution operator has no limit for adiabatic switchings, the Gell-Mann and Low formula allows to follow the evolution of eigenstates. In the degenerate case, for generic initial eigenstates, the adiabatic approximation (obtained by two different limiting procedures) is either useless or wrong, and the Gell-Mann and Low formula does not hold. We show how to select initial states in order to avoid such failures.

cond-mat.str-el↗

arXiv2008

X-ray Linear Dichroism in cubic compounds: the case of Cr3+ in MgAl2O4

The angular dependence (x-ray linear dichroism) of the Cr K pre-edge in MgAl2O4:Cr3+ spinel is measured by means of x-ray absorption near edge structure spectroscopy (XANES) and compared to calculations based on density functional theory (DFT) and ligand field multiplet theory (LFM). We also present an efficient method, based on symmetry considerations, to compute the dichroism of the cubic crystal starting from the dichroism of a single substitutional site. DFT shows that the electric dipole transitions do not contribute to the features visible in the pre-edge and provides a clear vision of the assignment of the 1s-->3d transitions. However, DFT is unable to reproduce quantitatively the angular dependence of the pre-edge, which is, on the other side, well reproduced by LFM calculations. The most relevant factors determining the dichroism of Cr K pre-edge are identified as the site distortion and 3d-3d electronic repulsion. From this combined DFT, LFM approach is concluded that when the pre-edge features are more intense than 4 % of the edge jump, pure quadrupole transitions cannot explain alone the origin of the pre-edge. Finally, the shape of the dichroic signal is more sensitive than the isotropic spectrum to the trigonal distortion of the substitutional site. This suggests the possibility to obtain quantitative information on site distortion from the x-ray linear dichroism by performing angular dependent measurements on single crystals.

cond-mat.mtrl-sci↗

arXiv2008

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.

cond-mat.str-el↗

arXiv2008

Relating on-shell and off-shell formalism in perturbative quantum field theory

In the on-shell formalism (mostly used in perturbative quantum field theory) the entries of the time ordered product T are on-shell fields (i.e. the basic fields satisfy the free field equations). With that, (multi)linearity of T is incompatible with the Action Ward identity. This can be circumvented by using the off-shell formalism in which the entries of T are off-shell fields. To relate on- and off-shell formalism correctly, a map sigma from on-shell fields to off-shell fields was introduced axiomatically by Duetsch and Fredenhagen. In that paper it was shown that, in the case of one real scalar field in N=4 dimensional Minkowski space, these axioms have a unique solution. However, this solution was given there only recursively. We solve this recurrence relation and give a fully explicit expression for sigma in the cases of the scalar, Dirac and gauge fields for arbitrary values of the dimension N.

hep-th↗