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Semiclassical measure of the propagation between two topological insulators

Éric Vacelet

TL;DR

This work analyzes the semiclassical propagation of quantum states at the interface between two topological insulators modeled by a Dirac equation with a spatially varying mass. By reducing the operator to a normal form near the interface and employing a two-scale Wigner framework in normal geodesic coordinates, it derives a complete description of the limiting measures: away from the interface the Wigner measure evolves under Liouville dynamics associated with $\lambda_\pm$, while near the interface it decomposes into a finite family of transverse modes and a two-scale measure on the normal bundle, both governed by explicit transport equations. The results precisely describe how mass concentrates along the interface and splits into transverse bands, providing a rigorous framework for edge-state dynamics and the fate of dispersive remnants in the $\varepsilon\to0$ limit, with potential implications for predicting edge transport in TI devices. Overall, the paper delivers a rigorous, two-scale, microlocal account of interface-guided propagation in Dirac-type models, bridging dispersive phenomena and edge-state transport in topological insulators.

Abstract

We study the propagation of initial condition in the presence of two topological insulators without magnetic field where the interface is a smooth connected not compact curve without boundaries. The solution is governed by an adiabatic modulation of a Dirac operator with a variable mass. We determine the evolution of the semiclassical measure of the solution with a two-scale Wigner measure method by reducing the Dirac operator to a normal form.

Semiclassical measure of the propagation between two topological insulators

TL;DR

This work analyzes the semiclassical propagation of quantum states at the interface between two topological insulators modeled by a Dirac equation with a spatially varying mass. By reducing the operator to a normal form near the interface and employing a two-scale Wigner framework in normal geodesic coordinates, it derives a complete description of the limiting measures: away from the interface the Wigner measure evolves under Liouville dynamics associated with , while near the interface it decomposes into a finite family of transverse modes and a two-scale measure on the normal bundle, both governed by explicit transport equations. The results precisely describe how mass concentrates along the interface and splits into transverse bands, providing a rigorous framework for edge-state dynamics and the fate of dispersive remnants in the limit, with potential implications for predicting edge transport in TI devices. Overall, the paper delivers a rigorous, two-scale, microlocal account of interface-guided propagation in Dirac-type models, bridging dispersive phenomena and edge-state transport in topological insulators.

Abstract

We study the propagation of initial condition in the presence of two topological insulators without magnetic field where the interface is a smooth connected not compact curve without boundaries. The solution is governed by an adiabatic modulation of a Dirac operator with a variable mass. We determine the evolution of the semiclassical measure of the solution with a two-scale Wigner measure method by reducing the Dirac operator to a normal form.

Paper Structure

This paper contains 31 sections, 19 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Topological insulators and Dirac type equations
  3. Edge states generated by coherent states
  4. Main result
  5. Main ideas of the proofs and organisation of the article
  6. Notation
  7. Semiclassical analysis
  8. Wigner transform and Wigner measure
  9. Application to the Dirac equation
  10. -oscillation of solution to Dirac equation
  11. Time-averaged Wigner measures
  12. First properties of the Wigner measure of the solution of the Dirac equation
  13. Two-scale analysis of concentrations
  14. Normal geodesic coordinates
  15. Two-scale Wigner measures
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\vec{f} \in \mathscr S \left(\mathbb{R}^2,\mathbb{C}^2 \right)$ independent of $\varepsilon$. If ${ \left(\psi_t^\varepsilon \right)_{\varepsilon > 0} }$ solves eqasol with for all $\varepsilon \in (0,1]$ and for all $x \in \mathbb{R}^2$, then there exists $T > 0$, such that for all $t \in (0,T)$, uniformly for $\varepsilon \in (0,1]$, where $\mathrm{WP}^\varepsilon_{x_t,0}$ is a wave packe

Theorems & Definitions (48)

  • Theorem 1.1: Edge-States Theorem 2 and Drouot Theorem A.1
  • Theorem 1.4
  • Remark : Link between Theorem \ref{['theodrouot']} and Theorem \ref{['theo:intro_gen']}
  • Theorem 1.5
  • Example 2.1
  • Lemma 2.2: Conservation of $\varepsilon$-oscillating
  • proof : Proof of Lemma \ref{['eposc']}
  • Lemma 2.3: Evolution of semiclassical measure outside $\mathcal{C}$
  • Proposition 3.1: Hamiltonian Normal form near the interface
  • Remark
  • ...and 38 more