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Algebraic Aspects of Periodic Graph Operators

Stephen P. Shipman, Frank Sottile

TL;DR

The paper develops an algebraic framework for periodic graph operators using the discrete $ Z^d$ action and Floquet theory, treating Bloch and Fermi varieties as algebraic objects that encode the link between translation eigenvalues and operator spectra. By viewing operators as $R$-module endomorphisms with $R=\mathbb{C}[z^{\pm}]$, it separates algebraic and analytic aspects of the spectrum, derives a full Floquet-Bloch spectral resolution on $\ell^2(\mathcal{V})$, and relates eigenfunctions, density of states, and spectral bands to the zero set $D(z,\lambda)$ of the dispersion function. The work emphasizes reducibility of Bloch/Fermi varieties via symmetry, multilayer constructions, and edge-contraction schemes, and explores defect modes within the continuum, illustrated by graphene stacking models. It then develops toric-geometry methods, via toric compactification and Newton polytopes, to study nondegeneracy and asymptotic spectral problems, offering a unified algebraic approach with broad implications for spectral theory, isospectrality, and defect phenomena in periodic graphs.

Abstract

A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, Bloch and Fermi varieties. These encode the relation between the eigenvalues of the translation group and the eigenvalues of the operator. As they are algebraic varieties, algebraic methods may be used to study the spectrum of the operator. We establish a framework in which commutative algebra directly comes to bear on the spectral theory of periodic operators, helping to distinguish their algebraic and analytic aspects. We also discuss reducibility of the Fermi variety and non-degeneracy of spectral band edges.

Algebraic Aspects of Periodic Graph Operators

TL;DR

The paper develops an algebraic framework for periodic graph operators using the discrete action and Floquet theory, treating Bloch and Fermi varieties as algebraic objects that encode the link between translation eigenvalues and operator spectra. By viewing operators as -module endomorphisms with , it separates algebraic and analytic aspects of the spectrum, derives a full Floquet-Bloch spectral resolution on , and relates eigenfunctions, density of states, and spectral bands to the zero set of the dispersion function. The work emphasizes reducibility of Bloch/Fermi varieties via symmetry, multilayer constructions, and edge-contraction schemes, and explores defect modes within the continuum, illustrated by graphene stacking models. It then develops toric-geometry methods, via toric compactification and Newton polytopes, to study nondegeneracy and asymptotic spectral problems, offering a unified algebraic approach with broad implications for spectral theory, isospectrality, and defect phenomena in periodic graphs.

Abstract

A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, Bloch and Fermi varieties. These encode the relation between the eigenvalues of the translation group and the eigenvalues of the operator. As they are algebraic varieties, algebraic methods may be used to study the spectrum of the operator. We establish a framework in which commutative algebra directly comes to bear on the spectral theory of periodic operators, helping to distinguish their algebraic and analytic aspects. We also discuss reducibility of the Fermi variety and non-degeneracy of spectral band edges.

Paper Structure

This paper contains 58 sections, 9 theorems, 115 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $\mathbb{Z}^d$ actions and the Fourier transform
  3. Algebraic structure
  4. Translation of vertices
  5. Translation of functions
  6. Formal Fourier transform
  7. $L^2$ Floquet transform and Fourier inversion
  8. Eigenfunctions of the translation group
  9. Fourier inversion
  10. Completion to $L^2$
  11. Spec and spectrum
  12. Periodic graph operators
  13. On $F_0({\mathcal{V}})$: Module endomorphism
  14. Definition and notation
  15. Examples
  16. ...and 43 more sections

Key Result

Theorem 2

A $\mathbb{C}$-linear operator $A$ on $F_0({\mathcal{V}})$ is periodic if and only if $\hat{A}$ is a module endomorphism of $\mathbb{C}[z^\pm]^W$.

Figures (3)

  • Figure 1: Some Bloch varieties of AB- and AA- stacked bi-layer graphene.
  • Figure 2: Filonov-Kachkovskiy Bloch variety with a curve of critical points.
  • Figure 3: Four Newton polytopes

Theorems & Definitions (12)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Theorem 6
  • Theorem 7
  • Theorem 8: DoKuchmentSottile2019a
  • ...and 2 more