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On the spectrum of magnetic Laplacian on the Lieb lattice

Moises Gomez Solis, Dylan Spedale, Fan Yang

Abstract

We study the magnetic Laplacian on the Lieb lattice, and prove Cantor spectrum for arbitrary irrational magnetic flux. We also provide a complete spectral analysis for the reduced one-dimensional Hamiltonian, proving Cantor spectra for all irrational frequencies, and sharp arithmetic phase transitions. Part of our analysis reveals a novel coexistence phenomenon of point spectrum and absolutely/singular continuous spectrum.

On the spectrum of magnetic Laplacian on the Lieb lattice

Abstract

We study the magnetic Laplacian on the Lieb lattice, and prove Cantor spectrum for arbitrary irrational magnetic flux. We also provide a complete spectral analysis for the reduced one-dimensional Hamiltonian, proving Cantor spectra for all irrational frequencies, and sharp arithmetic phase transitions. Part of our analysis reveals a novel coexistence phenomenon of point spectrum and absolutely/singular continuous spectrum.
Paper Structure (13 sections, 20 theorems, 60 equations, 2 figures)

This paper contains 13 sections, 20 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Main results
  4. Historical results on the almost Mathieu operator
  5. Cantor spectrum
  6. Spectral decompositions
  7. Proofs of theorems
  8. The zero energy: Theorem \ref{['thm:zero']}
  9. Reduction to the almost Mathieu operator
  10. Proofs of Theorems \ref{['thm:away_from_zero']} and \ref{['thm:Cantor_2d']}
  11. Proof of Theorem \ref{['thm:spectral_dec']}
  12. An alternate reduction from $H_{\alpha,t,\theta}$ to $H^{\mathrm{AMO}}_{\alpha,\theta,t^{-2}}$
  13. Lieb lattice with general coupling

Key Result

Theorem 1.1

For any irrational $\alpha$ and any $\theta\in \mathbb T$, $0\in \sigma(H_{\alpha,t,\theta})$.

Figures (2)

  • Figure 1: Lieb Lattice
  • Figure 2: Lieb Lattice

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • ...and 14 more