Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation

Abstract

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, increasing in precision, can be distinguished: the semiclassical theory given by the Vlasov equation, the mean-field theory given by the Hartree-Fock equation, and the description of the dominant effects of non-trivial entanglement by the random phase approximation. Particular attention is given to the discussion of admissible initial data, and I present an example of a realistic quantum quench that can be approximated by Hartree-Fock dynamics.

arXiv commentcontribution to the proceedings of ICMP 2021, v3: minor corrections

References in corpus (26)

• Mean-field Evolution of Fermionic Systems
• Mean field evolution of fermions with Coulomb interaction
• From the Hartree dynamics to the Vlasov equation
• Time evolution of the Luttinger model with nonuniform temperature profile
• Mean-Field Dynamics of Fermions with Relativistic Dispersion
• A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics
• Steady states and universal conductance in a quenched Luttinger model
• Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction
• The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
• Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime
• Correlation Energy of a Weakly Interacting Fermi Gas
• The classical limit of the time dependent Hartree-Fock equation. I. The Weyl symbol of the solution
• Mean-field limit and Semiclassical Expansion of a Quantum Particle System
• The dilute Fermi gas via Bogoliubov theory
• Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet
• Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation
• Accuracy of the time-dependent Hartree-Fock approximation for uncorrelated initial states
• Semiclassical limit to the Vlasov equation with inverse power law potentials
• Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics
• From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials
• Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
• The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin
• Interaction Corrections to Spin-Wave Theory in the Large-S Limit of the Quantum Heisenberg Ferromagnet
• Strong semiclassical limit from Hartree and Hartree-Fock to Vlasov-Poisson equation
• Bosonic Collective Excitations in Fermi Gases
• Free energy asymptotics of the quantum Heisenberg spin chain