Niels Benedikter

Updated December 20, 2024

Bosonization and Correlations in Fermionic Systems

In the last years we developed an approach for the description of quantum correlations in fermionic many-body systems by approximate bosonization of collective particle-hole excitations. We use this approach to compute the correlation energy of the Fermi gas in the mean-field scaling regime, justifying the random-phase approximation (RPA). Moreover we identify plasmons in the spectrum of the effective theory, and prove a Fock space norm approximation of the fermionic dynamics in terms of an effective bosonic evolution. Most recently, we computed the two-point correlation function in the RPA.

Spin Wave Theory

We study corrections to the free energy in the Quantum Heisenberg Ferromagnet due to remainder interaction effects in the Spin Wave Theory. We obtain a partial verification of Dyson's claim that spin-wave interactions have extremely small effects at low temperature.

Efficient Evaluation of Solid Harmonic Gaussian Integrals

We derive explicit formulas for certain integrals in numerical quantum chemistry calculations. The integral scheme is implemented in the CP2K quantum chemistry software, leading to a three orders of magnitude speed-up compared to previous methods.

Effective Evolution Equations

The following is a contribution to the proceedings of the ICMP 2021 in Geneva, reviewing the derivation of the different types of effective theories used to approximate fermionic many-body quantum systems: the classical approximation (Vlasov equation), the quantum mean-field theory keeping the fermionic kinematical correlations (Hartree-Fock theory), and the dominant dynamical correlations effects described by bosonization (random phase approximation).

We derive the fermionic Bogoliubov-de-Gennes equations (Hartree-Fock equations with pairing density; also called the time-dependent BCS equations, describing Cooper pairs in superconductivity) and the bosonic Hartree-Fock-Bogoliubov equations from a reformulation of the time-dependent variational principle, proving optimality of these approximations. We also give a proof of well-posedness for the Bogoliubov-de-Gennes equations with singular interactions.

In the following lecture notes we discuss a wide range of results concerning effective evolution equations for bosonic and fermionic systems.

The many-body Schrödinger equation in certain scaling regimes gives rise to effective nonlinear dynamics. An overview can be found in my thesis:

Fermionic Effective Evolution Equations

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 obtained in 2014 by Benedikter, Porta, and Schlein. First, the derivation holds in arbitrary space dimension. Second, by using an explicit formula for the unitary implementation of particle-hole transformations, we cast the proof in a form similar to the coherent state method of Rodnianski and Schlein for bosons.

We derive the time-dependent Hartree-Fock equation (TDHF) governing the effective dynamics of fermions in the mean-field regime. In a recent paper, we extend the derivation to mixed states as initial data, e.g., initial data prepared at positive temperature. As a second step of approximation, we derive the Vlasov equation of kinetic theory.

Bosonic Effective Evolution Equation

We derive the Gross-Pitaevskii equation describing the non-equilibrium properties of dilute Bose-Einstein condensates:

Quantum Electrodynamics

Physical experience shows that excited atoms relax to the ground state by emission of photons. We study the rate of relaxation in non-relativistic quantum electrodynamics: