arXiv subjects

Mathieu Lewin

Publications and source records attributed to Mathieu Lewin.

104 records · Page 6Linked to original sources

arXiv2008

A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case

This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock model by neglecting the exchange term. First, we recall the definition of the self-consistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in details a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, Séré and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the self-consistent Fermi sea in the presence of a defect. We end the paper by proving that our model is in fact the thermodynamic limit of the so-called supercell model, widely used in numerical simulations.

math-ph↗

arXiv2008

Properties of the periodic Hartree-Fock minimizer

We study the periodic Hartree-Fock model used for the description of electrons in a crystal. The existence of a minimizer was previously shown by Catto, Le Bris and Lions (Ann. Inst. H. Poincare Anal. Non Lineaire} 18 (2001), no.6, 687--760). We prove in this paper that any minimizer is necessarily a projector and that it solves a certain nonlinear equation, similarly to the atomic case. In particular we show that the Fermi level is either empty or totally filled.

math-ph↗

arXiv2008

A Nonlinear Model for Relativistic Electrons at Positive Temperature

We study the relativistic electron-positron field at positive temperature in the Hartree-Fock-approximation. We consider both the case with and without exchange term, and investigate the existence and properties of minimizers. Our approach is non-perturbative in the sense that the relevant electron subspace is determined in a self-consistent way. The present work is an extension of previous work by Hainzl, Lewin, Séré, and Solovej where the case of zero temperature was considered.

math-ph↗

arXiv2008

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of no-photon Quantum Electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of $N$ electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant $α$ is small whereas $αZ$ and the particle number $N$ are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently $α$ tends to zero) and $Z$, $N$ are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree-Fock ground state.

math-ph↗

arXiv2008

Stability of the Hartree-Fock model with temperature

This paper is devoted to the Hartree-Fock model with temperature in the euclidean space. For large classes of free energy functionals, minimizers are obtained as long as the total charge of the system does not exceed a threshold which depends on the temperature. The usual Hartree-Fock model is recovered in the zero temperature limit. An orbital stability result for the Cauchy problem is deduced from the variational approach.

math.AP↗

arXiv2008

Non-perturbative embedding of local defects in crystalline materials

We present a new variational model for computing the electronic first-order density matrix of a crystalline material in presence of a local defect. A natural way to obtain variational discretizations of this model is to expand the difference Q between the density matrix of the defective crystal and the density matrix of the perfect crystal, in a basis of precomputed maximally localized Wannier functions of the reference perfect crystal. This approach can be used within any semi-empirical or Density Functional Theory framework.

cond-mat.other↗

arXiv2008

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R^3, the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems.

math-ph↗

arXiv2007

Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms

We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant $α$. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an $L^1$ function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to $α$ by the formula $α_{\rm phys}\simeq α(1+2α/(3π)\logΛ)^{-1}$, where $Λ$ is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge $Z$. In the nonrelativistic limit, we obtain that this number is $\leq 2Z$, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Sere and Solovej on the mean-field approximation of no-photon QED.

math-ph↗

arXiv2007

A Minimization Method for Relativistic Electrons in a Mean-Field Approximation of Quantum Electrodynamics

We study a mean-field relativistic model which is able to describe both the behavior of finitely many spin-1/2 particles like electrons and of the Dirac sea which is self-consistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the photon field. All our results are non-perturbative and mathematically rigorous.

physics.atom-ph↗

arXiv2006

The Electronic Ground State Energy Problem: a New Reduced Density Matrix Approach

We present here a formulation of the electronic ground-state energy in terms of the second order reduced density matrix, using a duality argument. It is shown that the computation of the ground-state energy reduces to the search of the projection of some two-electron reduced Hamiltonian on the dual cone of $N$-representability conditions. Some numerical results validate the approach, both for equilibrium geometries and for the dissociation curve of N$_2$.

quant-ph↗

arXiv2005

The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case

We study the mean-field approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal-ordering or choice of bare electron/positron subspaces. Neglecting photons, we define properly this Hamiltonian in a finite box $[-L/2;L/2)^3$, with periodic boundary conditions and an ultraviolet cut-off $Λ$. We then study the limit of the ground state (i.e. the vacuum) energy and of the minimizers as $L$ goes to infinity, in the Hartree-Fock approximation. In case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as $L$ goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy.

math-ph↗

arXiv2005

Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation

According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator $D^0$. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane ({\it J. Phys. B}, 22, 3791--3814, 1989), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.

math-ph↗

arXiv2005

Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation

We consider a generalized Dirac-Fock type evolution equation deduced from no-photon Quantum Electrodynamics, which describes the self-consistent time-evolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently established by Hainzl-Lewin-Sere, we prove the existence of global-in-time solutions of the considered evolution equation.

math-ph↗

arXiv2005

Self-consistent solution for the polarized vacuum in a no-photon QED model

We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989) which is a mean-field theory deduced from no-photon QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a self-consistent equation. In a recent paper math-ph/0403005, we proved the convergence of the iterative fixed-point scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cut-off $Λ$ and the bare fine structure constant $α$. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off $Λ$ and without any constraint on the external field. We also study the behaviour of the minimizer as $Λ$ goes to infinity and show that the theory is "nullified" in that limit, as predicted first by Landau: the vacuum totally kills the external potential. Therefore the limit case of an infinite cut-off makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant $α$, on a simplified model where the exchange term is neglected.

physics.atom-ph↗