28 research outputs found
Robust determination of maximally-localized Wannier functions
We propose an algorithm to determine Maximally Localized Wannier Functions
(MLWFs). This algorithm, based on recent theoretical developments, does not
require any physical input such as initial guesses for the Wannier functions,
unlike popular schemes based on the projection method. We discuss how the
projection method can fail on fine grids when the initial guesses are too far
from MLWFs. We demonstrate that our algorithm is able to find localized Wannier
functions through tests on two-dimensional systems, simplified models of
semiconductors, and realistic DFT systems by interfacing with the Wannier90
code. We also test our algorithm on the Haldane and Kane-Mele models to examine
how it fails in the presence of topological obstructions
An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation
A perturbation-method-based <i>a posteriori</i> estimator for the planewave discretization of nonlinear Schrödinger equations
Renormalization and asymptotic expansion of Dirac's polarized vacuum
We perform rigorously the charge renormalization of the so-called reduced
Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac
operator, describes atoms and molecules while taking into account vacuum
polarization effects. We consider the total physical density including both the
external density of a nucleus and the self-consistent polarization of the Dirac
sea, but no `real' electron. We show that it admits an asymptotic expansion to
any order in powers of the physical coupling constant \alphaph, provided that
the ultraviolet cut-off behaves as \Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1.
The renormalization parameter $
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, S\'er\'e 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.Comment: Final version, to appear in Comm. Math. Phy
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 . 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 function and compute the
effective charge of the system, recovering the usual renormalization of charge:
the physical coupling constant is related to by the formula
, where
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 . In
the nonrelativistic limit, we obtain that this number is , 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.Comment: 37 pages, 1 figur