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The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials

Jinyeop Lee, Kunlun Qi

TL;DR

This work derives the semiclassical limit of the 2D Dirac--Hartree equation with a periodic background by projecting the matrix-valued Wigner function onto positive and negative energy bands. The authors show that the band-resolved densities $f_{\pm}^{\hbar}$ converge to solutions of relativistic Vlasov-type transport equations, with massive and massless regimes yielding distinct group velocities $\mathbf{v}(\mathbf{k})$; a Dirac-point cutoff is used in the massless case to handle interband coupling. The mean-field Hartree interaction and the periodic potential enter via a self-consistent potential, producing a coupled system for the limiting densities and total density $\rho$, with explicit error bounds of order $\mathcal{O}(\hbar)$ (up to $a^{-2}$ factors) in appropriate weak norms. As a corollary, the relativistic Vlasov--Poisson equation is recovered in the regularized Coulomb interaction setting when the regularization vanishes together with the semiclassical parameter. The results bridge quantum Dirac dynamics in periodic media (relevant to graphene-like systems) with classical relativistic transport, providing rigorous justification for relativistic mean-field transport in the semiclassical regime.

Abstract

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter.

The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials

TL;DR

This work derives the semiclassical limit of the 2D Dirac--Hartree equation with a periodic background by projecting the matrix-valued Wigner function onto positive and negative energy bands. The authors show that the band-resolved densities converge to solutions of relativistic Vlasov-type transport equations, with massive and massless regimes yielding distinct group velocities ; a Dirac-point cutoff is used in the massless case to handle interband coupling. The mean-field Hartree interaction and the periodic potential enter via a self-consistent potential, producing a coupled system for the limiting densities and total density , with explicit error bounds of order (up to factors) in appropriate weak norms. As a corollary, the relativistic Vlasov--Poisson equation is recovered in the regularized Coulomb interaction setting when the regularization vanishes together with the semiclassical parameter. The results bridge quantum Dirac dynamics in periodic media (relevant to graphene-like systems) with classical relativistic transport, providing rigorous justification for relativistic mean-field transport in the semiclassical regime.

Abstract

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter.
Paper Structure (18 sections, 12 theorems, 145 equations, 1 figure)

This paper contains 18 sections, 12 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model
  3. Background and motivation
  4. Organization of the paper
  5. Preliminaries
  6. Notation
  7. Wigner transform
  8. Positive and negative energy projection
  9. Main results
  10. Assumptions
  11. Main Theorems
  12. Proof of Theorems \ref{['thm:massiv']} and \ref{['thm:massless']}
  13. Estimate of the error term from the kinetic part
  14. Estimate of error term from external potential part
  15. Estimate of error term from interactive potential part
  16. ...and 3 more sections

Key Result

Lemma 2.2

Let $\psi^\hbar(t,\mathbf{x})\in L^2(\mathbb{R}^2;\mathbb{C}^2)$ and let $\mathcal{W}^\hbar(t,\mathbf{x},\mathbf{k})\in \mathbb{C}^{2\times 2}$ be the associated matrix-valued Wigner transform where $\overline{\psi} := \psi^\dagger \gamma^0$ is the Dirac adjoint. Then, the Dirac density function satisfies

Figures (1)

  • Figure 2: Diagram of the limiting process $\hbar \to 0$ and $c \to \infty$ for the massive case.

Theorems & Definitions (29)

  • Remark 2.1
  • Lemma 2.2: Spatial density as the $\mathbf{k}$-marginal of the Wigner transform
  • proof
  • Remark 2.3: Band decomposition of the spacial density
  • Remark 2.4: Physical interpretation of the two bands
  • Lemma 2.5: Derivatives of the spectral projectors
  • Remark 2.6
  • proof : Proof of Lemma \ref{['lem:grad-Pi']}
  • Theorem 3.1: Massive case ($m>0$)
  • Theorem 3.2: Massless case ($m=0$)
  • ...and 19 more