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On the magnetic perturbation theory for Chern insulators

Horia D. Cornean, Massimo Moscolari

TL;DR

This work develops a gauge covariant magnetic perturbation framework for Schrödinger operators with long-range magnetic perturbations, establishing resolvent-based perturbative control that yields stability of spectral gaps and precise behavior of the discrete spectrum under small magnetic changes. It introduces gauge-invariant constructions and approximate perturbed projections, proving strong (but not always norm) continuity of spectral projections and showing how spectral islands shift Lipschitzly with the magnetic field. The results illuminate how Chern insulators respond to magnetic perturbations, linking perturbation theory to topological invariants via the Fermi projection and Chern character, and providing practical tools for constructing field-dependent approximate projections and fixed-point expansions for isolated eigenvalues. A key implication is the rigorous description of bulk-edge transport phenomena in topological phases through gauge-covariant perturbation theory. Overall, the paper combines resolvent methods, Peierls phases, and Feshbach-type reductions to analyze magnetic perturbations in two-dimensional quantum systems with relevance to Chern insulators.

Abstract

The gauge covariant magnetic perturbation theory is tailored for one-body Schrödinger operators perturbed by long-range magnetic fields. In this work we present a self-contained exposition of the method, by outlining its technical foundations and discussing the physical heuristics behind the proofs. We apply it in order to prove the stability of spectral gaps and to study the location of the discrete spectrum. We also analyze the (lack of) continuity with respect to the magnetic field of spectral projections corresponding to finite spectral islands, which is a particularly important situation for systems modelling Chern insulators. Finally, we show how to construct approximate projections that have an explicit dependence with respect to the magnetic field parameter.

On the magnetic perturbation theory for Chern insulators

TL;DR

This work develops a gauge covariant magnetic perturbation framework for Schrödinger operators with long-range magnetic perturbations, establishing resolvent-based perturbative control that yields stability of spectral gaps and precise behavior of the discrete spectrum under small magnetic changes. It introduces gauge-invariant constructions and approximate perturbed projections, proving strong (but not always norm) continuity of spectral projections and showing how spectral islands shift Lipschitzly with the magnetic field. The results illuminate how Chern insulators respond to magnetic perturbations, linking perturbation theory to topological invariants via the Fermi projection and Chern character, and providing practical tools for constructing field-dependent approximate projections and fixed-point expansions for isolated eigenvalues. A key implication is the rigorous description of bulk-edge transport phenomena in topological phases through gauge-covariant perturbation theory. Overall, the paper combines resolvent methods, Peierls phases, and Feshbach-type reductions to analyze magnetic perturbations in two-dimensional quantum systems with relevance to Chern insulators.

Abstract

The gauge covariant magnetic perturbation theory is tailored for one-body Schrödinger operators perturbed by long-range magnetic fields. In this work we present a self-contained exposition of the method, by outlining its technical foundations and discussing the physical heuristics behind the proofs. We apply it in order to prove the stability of spectral gaps and to study the location of the discrete spectrum. We also analyze the (lack of) continuity with respect to the magnetic field of spectral projections corresponding to finite spectral islands, which is a particularly important situation for systems modelling Chern insulators. Finally, we show how to construct approximate projections that have an explicit dependence with respect to the magnetic field parameter.

Paper Structure

This paper contains 11 sections, 17 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Long-range magnetic perturbations
  3. Mathematical framework and results
  4. The quasi-inverse and stability of spectral gaps
  5. Some heuristics behind the gauge covariant magnetic perturbation theory
  6. Approximate perturbed projections
  7. Continuity in the strong topology and stability of the discrete spectrum
  8. Perturbation of discrete eigenvalues
  9. Lack of continuity in the norm topology
  10. Continuity and off-diagonal decay
  11. Integral kernel estimates for the resolvent

Key Result

Lemma 2.1

The family of magnetic vector potentials $\left\{A_{\bf y}: {\mathbb R}^2 \to {\mathbb R}^2\right\}_{{\bf y} \in {\mathbb R}^2}$ generates the same magnetic field $\mathfrak{b}({\bf x})$ As a consequence, the magnetic potentials $\left\{{A}_{\bf y}\right\}_{{\bf y} \in {\mathbb R}^2}$ differ by a pure gauge factor, namely there exists $\varphi_{\bf y}({\bf x}) \equiv \varphi({\bf x},{\bf y})$ such

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5: Cornean2010
  • Lemma 3.1: Simon1982
  • ...and 24 more