23 research outputs found
Bosonic Collective Excitations in Fermi Gases
Hartree-Fock theory has been justified as a mean-field approximation for
fermionic systems. However, it suffers from some defects in predicting physical
properties, making necessary a theory of quantum correlations. Recently,
bosonization of many-body correlations has been rigorously justified as an
upper bound on the correlation energy at high density with weak interactions.
We review the bosonic approximation, deriving an effective Hamiltonian. We then
show that for systems with Coulomb interaction this effective theory predicts
collective excitations (plasmons) in accordance with the random phase
approximation of Bohm and Pines, and with experimental observation.Comment: contribution to the proceedings of QMath14; Section 2 is a review of
arXiv:1809.01902, Section 3 is new material; v2: references update
Deriving the Gross-Pitaevskii equation
In experiments, Bose-Einstein condensates are prepared by cooling a dilute
Bose gas in a trap. After the phase transition has been reached, the trap is
switched off and the evolution of the condensate observed. The evolution is
macroscopically described by the Gross-Pitaevskii equation. On the microscopic
level, the dynamics of Bose gases are described by the -body Schr\"odinger
equation. We review our article [BdS12] in which we construct a class of
initial data in Fock space which are energetically close to the ground state
and prove that their evolution approximately follows the Gross-Pitaevskii
equation. The key idea is to model two-particle correlations with a Bogoliubov
transformation.Comment: 5 pages; contribution to the proceedings of QMath1
Effective Evolution Equations from Many-Body Quantum Mechanics
Systems of interest in physics often consist of a very large number of interacting particles. In certain physical regimes, effective non-linear evolution equations are commonly used as an approximation for making predictions about the time-evolution of such systems. Important examples are Bose-Einstein condensates of dilute Bose gases and degenerate Fermi gases. While the effective equations are well-known in physics, a rigorous justification is very difficult. However, a rigorous derivation is essential to precisely understand the range and the limits of validity and the quality of the approximation. In this thesis, we prove that the time evolution of Bose-Einstein condensates in the Gross-Pitaevskii regime can be approximated by the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation. We then turn to fermionic systems and prove that the evolution of a degenerate Fermi gas can be approximated by the time-dependent Hartree-Fock equation (TDHF) under certain assumptions on the semiclassical structure of the initial data. Finally, we extend the latter result to fermions with relativistic kinetic energy. All our results provide explicit bounds on the error as the number of particles becomes large. A crucial methodical insight on bosonic systems is that correlations can be modeled by Bogoliubov transformations. We construct initial data appropriate for the Gross-Pitaevskii regime using a Bogoliubov transformation acting on a coherent state, which amounts to studying squeezed coherent states. As a crucial insight for fermionic systems, we point out a semiclassical structure in states close to the ground state of fermions in a trap. As a convenient language for studying the dynamics of fermionic systems, we use particle-hole transformations
The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
The derivation of effective evolution equations is central to the study of
non-stationary quantum many-body sytems, and widely used in contexts such as
superconductivity, nuclear physics, Bose-Einstein condensation and quantum
chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of
reduced density matrices, and apply it to fermionic and bosonic many-body
systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov
equations, respectively. While we do not prove quantitative error estimates,
our formulation does show that the approximation is optimal within the class of
quasifree states. Furthermore, we prove well-posedness of the
Bogoliubov-de-Gennes equations in energy space and discuss conserved
quantities.Comment: 46 pages, 1 figure; v2: simplified proof of conservation of particle
number, additional references; v3: minor clarification
Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals
An integral scheme for the efficient evaluation of two-center integrals over
contracted solid harmonic Gaussian functions is presented. Integral expressions
are derived for local operators that depend on the position vector of one of
the two Gaussian centers. These expressions are then used to derive the formula
for three-index overlap integrals where two of the three Gaussians are located
at the same center. The efficient evaluation of the latter is essential for
local resolution-of-the-identity techniques that employ an overlap metric. We
compare the performance of our integral scheme to the widely used Cartesian
Gaussian-based method of Obara and Saika (OS). Non-local interaction potentials
such as standard Coulomb, modified Coulomb and Gaussian-type operators, that
occur in range-separated hybrid functionals, are also included in the
performance tests. The speed-up with respect to the OS scheme is up to three
orders of magnitude for both, integrals and their derivatives. In particular,
our method is increasingly efficient for large angular momenta and highly
contracted basis sets.Comment: 18 pages, 2 figures; accepted manuscript. v2: supplementary material
include
Mean-field evolution of fermions with singular interaction
We consider a system of N fermions in the mean-field regime interacting
though an inverse power law potential , for
. We prove the convergence of a solution of the many-body
Schr\"{o}dinger equation to a solution of the time-dependent Hartree-Fock
equation in the sense of reduced density matrices. We stress the dependence on
the singularity of the potential in the regularity of the initial data. The
proof is an adaptation of [22], where the case is treated.Comment: 16 page
Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential
Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only . Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy
Correlation Energy of a Weakly Interacting Fermi Gas
We derive rigorously the leading order of the correlation energy of a Fermi
gas in a scaling regime of high density and weak interaction. The result
verifies the prediction of the random-phase approximation. Our proof refines
the method of collective bosonization in three dimensions. We approximately
diagonalize an effective Hamiltonian describing approximately bosonic
collective excitations around the Hartree-Fock state, while showing that
gapless and non-collective excitations have only a negligible effect on the
ground state energy.Comment: 56 pages, 2 figures; v2: additional references on lattice point
countin
Correlation energy of a weakly interacting Fermi gas
We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree–Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy