Irreducibility of the Fermi Surface for Planar Periodic Graph Operators

TL;DR

This paper proves that for certain planar periodic graphs, the Fermi surface remains irreducible at most energies, with specific exceptions, impacting the spectral analysis of these operators.

Contribution

It establishes irreducibility of the Fermi surface for a broad class of planar periodic graph operators, including weighted Laplacians, under specific conditions.

Findings

Fermi surface is irreducible at all but finitely many energies for certain graphs.

Reducibility occurs only in special cases like the tetrakis square tiling with complex coefficients.

Explicit parameterization of reducible cases when coefficients are real.

Abstract

We prove that the Fermi surface of a connected doubly periodic self-adjoint discrete graph operator is irreducible at all but finitely many energies provided that the graph (1) can be drawn in the plane without crossing edges (2) has positive coupling coefficients (3) has two vertices per period. If "positive" is relaxed to "complex", the only cases of reducible Fermi surface occur for the graph of the tetrakis square tiling, and these can be explicitly parameterized when the coupling coefficients are real. The irreducibility result applies to weighted graph Laplacians with positive weights.