For three-dimensional interacting fermions at zero temperature, the ground-state Wigner function approaches Thomas-Fermi theory in trace norm at rate N times a positive power of the semi-classical parameter, with Coulomb interactions allowed.
Derivation of Hartree-Fock Dynamics and Semiclassical Commutator Estimates for Fermions in a Magnetic Field
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abstract
We study the quantum dynamics of a large number of interacting fermionic particles in a constant magnetic field. In a coupled mean-field and semiclassical scaling limit, we show that solutions of the many-body Schr\"odinger equation converge to solutions of a non-linear Hartree-Fock equation. The central ingredient of the proof are certain semiclassical trace norm estimates of commutators of the position and momentum operators with the one-particle density matrix of the solution of the Hartree-Fock equation. In a first step, we prove their validity for non-interacting initial data in a magnetic field by generalizing a 2020 result of Fournais and Mikkelsen. We then propagate these bounds from the initial data along the Hartree-Fock flow to arbitrary times.
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The quantitative semi-classical limit of a large Fermi system at zero temperature
For three-dimensional interacting fermions at zero temperature, the ground-state Wigner function approaches Thomas-Fermi theory in trace norm at rate N times a positive power of the semi-classical parameter, with Coulomb interactions allowed.