Algebraic Properties of the Fermi Variety for Periodic Graph Operators

TL;DR

This paper introduces a method to estimate and bound the number of irreducible components of Fermi varieties for periodic graph operators, linking algebraic properties to asymptotic analysis, with applications to various lattice models.

Contribution

It provides an abstract bound for the irreducibility of Laurent polynomials related to Fermi varieties, applicable to complex lattice structures.

Findings

Bound on the number of irreducible components derived

Irreducibility established for multiple lattice types

Applicable to decorated and multi-vertex fundamental cells

Abstract

We present a method to estimate the number of irreducible components of the Fermi varieties of periodic Schr\"odinger operators on graphs in terms of suitable asymptotics. Our main theorem is an abstract bound for the number of irreducible components of Laurent polynomials in terms of such asymptotics. We then show how the abstract bound implies irreducibility in many lattices of interest, including examples with more than one vertex in the fundamental cell such as the Lieb lattice as well as certain models obtained by the process of graph decoration.