## 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.

- N. Benedikter, S. Lill: Momentum Distribution of a Fermi Gas in the Random Phase Approximation, arXiv:2310.02706 [math-ph]
- N. Benedikter, C. Boccato: Correlation Corrections as a Perturbation to the Quasi-Free Approximation in Many-Body Quantum Systems, chapter in the Encyclopedia of Complexity and Systems Science, Robert A. Meyers (Ed), Springer.
- N. Benedikter, M. Porta, B. Schlein, and R. Seiringer: Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential, to appear in
*Archive for Rational Mechanics and Analysis*(2023) - N. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer: Bosonization of Fermionic Many-Body Dynamics,
*Annales Henri Poincaré***23**, 1725-1764 (2022) - N. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer: Correlation Energy of a Weakly Interacting Fermi Gas,
*Inventiones Mathematicae***225**(3), 885-979 (2021) - N. Benedikter: Bosonic Collective Excitations in Fermi Gases,
*Reviews in Mathematical Physics***32**, 2060009, (2020) - N. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer: Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime,
*Communications in Mathematical Physics***374**, 2097-2150 (2020)

## 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.

- N. Benedikter:
Interaction Corrections to Spin-Wave Theory in the in the Large-S Limit of the Quantum Heisenberg Ferromagnet,
*Mathematical Physics, Analysis, and Geometry***20**, 1-21 (2017)

## 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.

- D. Golze, N. Benedikter, M. Iannuzzi, J. Wilhelm, and J. Hutter:
Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals,
*Journal of Chemical Physics***146**, 034105 (2017)

## 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).

- N. Benedikter: Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation, arXiv:2203.11255 [math-ph] (2022)

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.

- N. Benedikter, J. Sok, and J.P. Solovej:
The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations,
*Annales Henri Poincaré*,**19**(4), 1167-1214 (2018)

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

- N. Benedikter, M. Porta, and B. Schlein:
Effective Evolution Equations from Quantum Dynamics (2016), in
*SpringerBriefs in Mathematical Physics*

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

- N. Benedikter: Effective Evolution Equations from Many-Body Quantum Mechanics, (2014) Thesis University of Bonn

## 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.

- N. Benedikter, D. Desio: Two Comments on the Derivation of the Time-Dependent Hartree-Fock Equation, contribution to the Proceedings of the Intensive Period "INdAM Quantum Meetings (IQM22)" at Politecnico di Milano, March - May 2022 (2022)

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.

- N. Benedikter, M. Porta. C. Saffirio, and B. Schlein: From the Hartree dynamics to the Vlasov equation,
*ARMA***221**, 273-334 (2016) - N. Benedikter, V. Jaksic, M. Porta, C. Saffirio, and B. Schlein: Mean-field Evolution of Fermionic Mixed States,
*Comm. Pure Appl. Math.***69**, 2250-2303 (2014) - N. Benedikter, M. Porta, and B. Schlein: Hartree-Fock dynamics for weakly interacting fermions (2014), in
*Proceedings of the QMath12 Conference* - N. Benedikter, M. Porta, and B. Schlein: Mean-Field Dynamics of Fermions with Relativistic Dispersion,
*J. Math. Phys.***55**, 021901 (2014) - N. Benedikter, M. Porta, and B. Schlein: Mean-field Evolution of Fermionic Systems,
*Comm. Math. Phys.***331**, 1087-1131 (2014)

## Bosonic Effective Evolution Equation

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

- N. Benedikter: Deriving the Gross-Pitaevskii Equation (2014), in
*Proceedings of the QMath12 Conference* - N. Benedikter, G. de Oliveira, and B. Schlein: Quantitative Derivation of the Gross-Pitaevskii Equation,
*Comm. Pure Appl. Math.***68**, 1399-1482 (2014)

## 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:

- N. Benedikter: Dynamics of the Radiative Decay of Excited Atoms (2010), Diploma thesis at the University of Stuttgart