arXiv subjects

Christian Brouder

Publications and source records attributed to Christian Brouder.

At least 55 records · Page 3Linked to original sources

arXiv2007

The structure of Green functions in quantum field theory with a general state

In quantum field theory, the Green function is usually calculated as the expectation value of the time-ordered product of fields over the vacuum. In some cases, especially in degenerate systems, expectation values over general states are required. The corresponding Green functions are essentially more complex than in the vacuum, because they cannot be written in terms of standard Feynman diagrams. Here, a method is proposed to determine the structure of these Green functions and to derive nonperturbative equations for them. The main idea is to transform the cumulants describing correlations into interaction terms.

hep-th↗

arXiv2007

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)^+), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of B are not renormalised, i.e. when Feynman diagrams containing one single vertex are not renormalised. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)^+) and the Faa di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra of diffeomorphisms is given. Finally, the bialgebra S(S(B)^+) is shown to give the same results as the standard renormalisation procedure for the scalar field.

hep-th↗

arXiv2007

Exponential localization of Wannier functions in insulators

The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.

cond-mat.mtrl-sci↗

arXiv2006

Analytical solutions for two-level systems with damping

A method is proposed to transform any analytic solution of the Bloch equation into an analytic solution of the Landau-Lifshitz-Gilbert equation. This allows for the analytical description of the dynamics of a two level system with damping. This method shows that damping turns the linear Schrödinger equation of a two-level system into a nonlinear Schrödinger equation. As applications, it is shown that damping has a relatively mild influence on self-induced transparency but destroys dynamical localization.

quant-ph↗

arXiv2006

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.

cond-mat.str-el↗

arXiv2005

Coalgebras and quantization

Two coalgebra structures are used in quantum field theory. The first one is the coalgebra part of a Hopf algebra leading to deformation quantization. The second one is a co-module co-algebra over the first Hopf algebra and it is used to define connected chronological products and renormalization.

math-ph↗

arXiv2005

Boundary conditions for augmented plane wave methods

The augmented plane wave method uses the Rayleigh-Ritz principle for basis functions that are continuous but with discontinuous derivatives and the kinetic energy is written as a pair of gradients rather than as a Laplacian. It is shown here that this procedure is fully justified from the mathematical point of view. The domain of the self-adjoint Hamiltonian, which does not contain functions with discontinuous derivatives, is extended to its form domain, which contains them, and this modifies the form of the kinetic energy. Moreover, it is argued that discontinuous basis functions should be avoided.

cond-mat.other↗

arXiv2005

Structural relaxations around Ti, Cr and Fe impurities in alpha-Al2O3 probed by x-ray absorption near edge structure combined with first-principles calculations

We determine the structural relaxations around paramagnetic impurities (Ti, Cr, Fe) in corundum (alpha-Al2O3), by combining x-ray absorption near edge structure (XANES) experiments and ab initio calculations. The structural relaxations are found to be very local. We then show that XANES is sensitive to small variations in interatomic distances within the coordination shell of the absorbing atom. The experiments were carried out on single crystals of ruby and sapphires. Linear dichroic signals are essential to characterize the geometry of the impurity site. The calculations were performed within a self-consistent ``non muffin-tin'' framework, that uses pseudopotentials, plane-wave basis set, and the continued fraction for the absorption cross section.

cond-mat.mtrl-sci↗

arXiv2004

Non-commutative Hopf algebra of formal diffeomorphisms

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the multiplication, while the second group is the set of formal diffeomorphisms with the composition. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization in quantum field theory, and to the renormalization functor given by the double tensor algebra on a bi-algebra.

math.QA↗

arXiv2004

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.

hep-th↗

arXiv2003

Many-body theory of degenerate systems

The many-body theory of degenerate systems is described in detail. More generally, this theory applies to systems with an initial state that cannot be described by a single Slater determinant. The double-source (or closed-time-path) formalism of nonequilibrium quantum field theory is used to derive an expression for the average value of a product of interacting fields when the initial state is not the vacuum or a Slater determinant. Quantum group techniques are applied to derive the hierarchy of unconnected Green functions and the hierarchy of connected ones.

cond-mat.str-el↗

arXiv2003

Many-body theory of degenerate systems: A simple example

The hierarchy of Green functions for (quasi)degenerate systems, presented in cond-mat/0308058, is calculated in detail for the case of a system with closed shells plus a single electron in a two-fold degenerate level. The complete hierarchy is derived explicitly, in terms of Green functions and of Feynman diagrams.

cond-mat.str-el↗

arXiv2003

A many-body approach to crystal field theory

A self-consistent many-body approach is proposed to build a first-principles crystal field theory, where crystal field parameters are calculated ab initio. Many-body theory is used to write the energy of the interacting system as a function of the density matrix of the noninteracting system. A variation of the energy with respect to the density matrix gives an effective Hamiltonian that is diagonalized to determine the density matrix providing the lowest energy. The equations are written in terms of the Hopf algebra of functional derivatives with respect to external fermionic sources. This approach contains the many-body theory of Green functions as a special case, and the usual crystal field theory as a first approximation. Therefore, it is expected to provide good results for strongly-interacting electron systems.

cond-mat.str-el↗

arXiv2003

Quantum field theory of degenerate systems

To set up a self-consistent quantum field theory of degenerate systems, the unperturbed state should be described by a density matrix instead of a pure state. This increases the combinatorial complexity of the many-body equations. Hopf algebraic techniques are used to deal with this complexity and show that the Schwinger-Dyson equations are modified in a non-trivial way. The hierarchy of Green functions is derived for degenerate systems, and the case of a single electron in a two-fold degenerate orbital is calculated in detail.

hep-th↗

arXiv2002

Quantum groups and interacting quantum fields

If C is a cocommutative coalgebra, a bialgebra structure can be given to the symmetric algebra S(C). The symmetric product is twisted by a Laplace pairing and the twisted product of any number of elements of S(C) is calculated explicitly. This is used to recover important identities in the quantum field theory of interacting scalar bosons.

hep-th↗

arXiv2002

Quantum groups and quantum field theory: I. The free scalar field

The quantum field algebra of real scalar fields is shown to be an example of infinite dimensional quantum group. The underlying Hopf algebra is the symmetric algebra S(V) and the product is Wick's normal product. Two coquasitriangular structures can be built from the two-point function and the Feynman propagator of scalar fields to reproduce the operator product and the time-ordered product as twist deformations of the normal product. A correspondence is established between the quantum group and the quantum field concepts. On the mathematical side the underlying structures come out of Hopf algebra cohomology.

hep-th↗

arXiv2002

Quantum fields and quantum groups

Quantum fields are shown to provide an example of infinite-dimensional quantum groups. A dictionary is established between quantum field and quantum group concepts: the expectation value over the vacuum is the counit, Wick's theorem is the definition of a twisted product, operator and time-ordered products are examples of twisted products. Through this dictionary, coquasitriangular structures are introduced in quantum field theory. These structures are the origin of Wick's theorem and quasifree states. Renormalization becomes the replacement of a coquasitriangular structure by a 2-coboundary. Quantum groups provide a second quantization without commutators which can second-quantize noncommutative algebras.

hep-ph↗

arXiv2002

Multiple plasmon resonances in naturally-occurring multiwall nanotubes: infrared spectra of chrysotile asbestos

Chrysotile asbestos is formed by densely packed bundles of multiwall hollow nanotubes. Each wall in the nanotubes is a cylindrically wrapped layer of $Mg_3 Si_2 O_5 (OH)_4$. We show by experiment and theory that the infrared spectrum of chrysotile presents multiple plasmon resonances in the Si-O stretching bands. These collective charge excitations are universal features of the nanotubes that are obtained by cylindrically wrapping an anisotropic material. The multiple plasmons can be observed if the width of the resonances is sufficiently small as in chrysotile.

cond-mat.mtrl-sci↗