We show how to derive an effective nonlinear dynamics, described by the Hartree-Fock equations, for fermionic quantum particles confined to a two-dimensional box and in presence of an external, uniform magnetic field. The derivation invokes the Dirac-Frenkel principle. We discuss the validity of this effective description with respect to the many-body Schrödinger dynamics for small times and for weak interactions, and also in regards to the number of particles.

• 35Q55 - NLS equations (nonlinear Schrödinger equations)
• 81Q10 - Selfadjoint operator theory in quantum theory, including spectral analysis
• 81V70 - Many-body theory; quantum Hall effect