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Derivation of Hartree-Fock Dynamics and Semiclassical Commutator Estimates for Fermions in a Magnetic Field

Niels Benedikter, Chiara Boccato, Domenico Monaco, Ngoc Nhi Nguyen

TL;DR

This work proves that a large, interacting fermion system in a constant magnetic field evolves, in a coupled mean-field and semiclassical regime, toward a nonlinear time-dependent Hartree–Fock equation. The authors develop sharp semiclassical trace-norm bounds for commutators of spectral projectors with position and momentum in the magnetic setting, and they propagate these bounds along the HF flow. They then derive rigorous HF dynamics from the underlying many-body Schrödinger dynamics, providing explicit, exponential-in-time error bounds in trace and Hilbert–Schmidt norms. The results rely on magnetic Weyl laws, local-to-global commutator analyses, and Agmon/CLR techniques to handle classically forbidden regions, and they extend prior non-magnetic results to the magnetized case with careful control of the magnetic field on the semiclassical scale. Overall, the paper delivers a rigorous pathway from many-body quantum dynamics of fermions in a magnetic field to the time-dependent Hartree–Fock description, with explicit quantitative estimates useful for further mathematical and physical applications.

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 Schrödinger 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.

Derivation of Hartree-Fock Dynamics and Semiclassical Commutator Estimates for Fermions in a Magnetic Field

TL;DR

This work proves that a large, interacting fermion system in a constant magnetic field evolves, in a coupled mean-field and semiclassical regime, toward a nonlinear time-dependent Hartree–Fock equation. The authors develop sharp semiclassical trace-norm bounds for commutators of spectral projectors with position and momentum in the magnetic setting, and they propagate these bounds along the HF flow. They then derive rigorous HF dynamics from the underlying many-body Schrödinger dynamics, providing explicit, exponential-in-time error bounds in trace and Hilbert–Schmidt norms. The results rely on magnetic Weyl laws, local-to-global commutator analyses, and Agmon/CLR techniques to handle classically forbidden regions, and they extend prior non-magnetic results to the magnetized case with careful control of the magnetic field on the semiclassical scale. Overall, the paper delivers a rigorous pathway from many-body quantum dynamics of fermions in a magnetic field to the time-dependent Hartree–Fock description, with explicit quantitative estimates useful for further mathematical and physical applications.

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 Schrödinger 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.

Paper Structure

This paper contains 16 sections, 24 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. First Main Result: Semiclassical Structure of Spectral Projections
  4. Commutator Estimates: State of the Art and Applications
  5. Second Main Result: Validity of the Hartree--Fock Evolution
  6. Semiclassical Structures for Spectral Projectors and Proof of \ref{['thm:cb-princ']}
  7. The Local Case
  8. Preliminaries and auxiliary local bounds
  9. Magnetic Weyl laws
  10. Local estimates with non-criticality condition
  11. Auxiliary Lemmata for the Classically Forbidden Region
  12. Proof of \ref{['thm:cb-princ']}
  13. Proof of the Propagation of the Semiclassical Structure (\ref{['thm:propag-semicl-magn']})
  14. Derivation of the Time-Dependent Hartree--Fock Equation (\ref{['thm:hf']})
  15. Spectral Theory of Magnetic Schrödinger Operators
  16. ...and 1 more sections

Key Result

Theorem 1

Let $a:\mathbb{R}^d\to\mathbb{R}^d$, $V:\mathbb{R}^d\to\mathbb{R}$ and $\mu\in\mathbb{R}$ satisfy ass:a-V. Let $C_0>0$ and $h_0>0$, let $P_j:=-i\hbar\partial_{x_j}-b a_j(x)$ be the $j$-th coordinate of the magnetic one–particle momentum operator $P=-i\hbar\nabla-b a(x)$ for $j\in\{1,\ldots,d\}$, and Then, there exists $C>0$ such that, for any $h\in(0,h_0]$, for any $b\in(0,C_0\hbar^{-1})$, and for

Theorems & Definitions (45)

  • Theorem 1: Semiclassical Structure of Initial Data
  • Remark 1.3
  • Theorem 2: Propagation of the Semiclassical Structure
  • Remark 1.4
  • Theorem 3: Validity of the Hartree--Fock Equation in a Magnetic Field
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.2: Local magnetic commutators bounds
  • Lemma 2.3: Diamagnetic inequality, Lieb-Seiringer-2010Stability
  • Lemma 2.4
  • ...and 35 more