Spectral gaps and discrete magnetic Laplacians

TL;DR

This paper establishes a geometric condition for spectral gaps in the discrete Laplacian on periodic graphs by analyzing the magnetic Laplacian and applying a virtualisation technique, with applications to materials like polypropylene.

Contribution

It introduces a simple geometric criterion for spectral gaps and develops a virtualisation method to analyze the magnetic Laplacian on periodic graphs.

Findings

Proves Higuchi-Shirai's conjecture for Z-periodic trees.

Demonstrates spectral gaps in examples like polypropylene and polyacetylene.

Provides a new approach to localize the spectrum of the Laplacian.

Abstract

The aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on the finite quotient and interpret the vector potential as a Floquet parameter. We develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues (written in ascending order and counting multiplicities) specify the bracketing intervals where the spectrum of the Laplacian is localised. We prove Higuchi-Shirai's conjecture for Z-periodic trees and apply our technique in several examples like the polypropylene or the polyacetylene to show the existence spectral gaps.