Spectral gaps and discrete magnetic Laplacians

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 of spectral gaps.2010 Mathematics Subject Classification. 05C50, 47B39, 47A10, 05C65.

Covering Graphs, Magnetic Spectral Gaps and Applications to Polymers and Nanoribbons

“…In this article, we generalize the geometric condition obtained in ( [18], Theorem 4.4) for β = 0 to non-trivial periodic magnetic potentials. In particular, if W = ( G, m) is a Γ-periodic MW-graph with magnetic potential β, we will give in Theorem 3 a simple geometric condition on the quotient graph G = G/Γ that guarantees the existence of non-trivial spectral gaps on the spectrum of the discrete magnetic Laplacian ∆ W β .…”

“…In this section, we introduce the basic definitions and results concerning MW-graphs and also define discrete magnetic Laplacians. For further motivation and results, we refer to [12,18,19] and references cited therein.…”

“…To show the existence of spectral gaps, we develop a purely discrete spectral localization technique based on the virtualization of arcs and vertices on quotient G. These operations produce new graphs with, in general, different weights that allow localizing the eigenvalues of the original Laplacian inside certain intervals. We call this procedure discrete bracketing, and we refer to [18] for additional motivation and proofs.…”

“…One of the new aspects of the present article is the generalization of results in [18] to include a periodic magnetic field β on the covering graph π : G → G = G/Γ. In this sense, β may be used as a control parameter for the system that serves to modify the size and the regions where the spectral gaps are localized.…”

“…Moreover, we will present the basic arc and vertex virtualization procedure that will allow one to localize the spectrum of the DML on the infinite covering graph. In Section 4, we extend the discrete Floquet theory considered in ( [18], Section 5) to the case of covering graphs with periodic magnetic potentials. In Section 5, we apply the spectral localization results developed before in the example of Z-periodic graphs modeling the polyacetylene polymer as well as graphene nanoribbons in the presence of a constant magnetic field.…”