Bosonization of Fermionic Many-Body Dynamics

  title={Bosonization of Fermionic Many-Body Dynamics},
  author={Niels Benedikter and Phan Th{\`a}nh Nam and Marcello Porta and Benjamin Schlein and Robert Seiringer},
  journal={Annales Henri Poincar{\'e}},
We consider the quantum many–body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing collective particle–hole pair excitations on the Fermi ball. Using a rigorous version of approximate bosonization, we prove that the many–body evolution can be approximated in Fock space norm by a quasifree bosonic evolution of the collective particle–hole excitations. 

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

Effective Dynamics of Extended Fermi Gases in the High-Density Regime

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus

Effective dynamics of interacting fermions from semiclassical theory to the random phase approximation

I review results concerning the derivation of effective equations for the dynamics of interacting Fermi gases in a high-density regime of mean-field type. Three levels of effective theories,

On the Effective Quasi-Bosonic Hamiltonian of the Electron Gas: Collective Excitations and Plasmon Modes

We consider an effective quasi-bosonic Hamiltonian of the electron gas which emerges naturally from the random phase approximation and describes the collective excitations of the gas. By a rigorous

Bogoliubov theory for the dilute Fermi gas in three dimensions

In a dilute system of N fermions with spin 1 / 2 in three dimensions, we study the correlation energy which is given by the difference between the ground state energy and the energy of the

The Gell-Mann$-$Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime

We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in

Two Comments on the Derivation of the Time-Dependent Hartree-Fock Equation

We revisit the derivation of the time–dependent Hartree–Fock equation for interacting fermions in a regime coupling a mean–field and a semiclassical scaling, contributing two comments to the result



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

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

Mean Field Evolution of Fermions with Coulomb Interaction

We study the many body Schrödinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges

On the Correlation Energy of Interacting Fermionic Systems in the Mean-Field Regime

We consider a system of $$N\gg 1$$ N ≫ 1 interacting fermionic particles in three dimensions, confined in a periodic box of volume 1, in the mean-field scaling. We assume that the interaction

Mean-field dynamics of fermions with relativistic dispersion

We extend the derivation of the time-dependent Hartree-Fock equation recently obtained by Benedikter et al. [“Mean-field evolution of fermionic systems,” Commun. Math. Phys. (to be published)] to

Mean–Field Evolution of Fermionic Systems

The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems.

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

This paper starts from the Hartree–Fock state given by plane waves and introduces collective particle–hole pair excitations, and uses Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy.

Fluctuations around Hartree states in the mean-field regime

We consider the dynamics of a large system of $N$ interacting bosons in the mean-field regime where the interaction is of order $1/N$. We prove that the fluctuations around the nonlinear Hartree

Fluctuations of N-particle quantum dynamics around the nonlinear Schrödinger equation


We consider some two-body operators acting on a Fock space with either fermionic or no statistics. We prove that they are bounded below by one-body operators which mimic exchange effects. This allows