Irreducibility of the Fermi Surface for Planar Periodic Graph Operators

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.

References in corpus (2)

• Generic properties of dispersion relations for discrete periodic operators
• Reducible Fermi surface for multi-layer quantum graphs including stacked graphene

Cited by in corpus (4)

• Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues
• Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators
• Topics on Fermi varieties of discrete periodic Schrödinger operators
• Fermi isospectrality of discrete periodic Schrödinger operators with separable potentials on ${\mathbb{Z}}^2$