Rare Flat Bands for Periodic Graph Operators

TL;DR

The paper proves that flat bands are generically absent for connected ${\mathbb Z}^d$-periodic graphs when edge weights and potentials are treated as variables. The approach combines Floquet theory with algebraic geometry, using dispersion polynomials $D(z,\lambda)$, Newton polytopes, and facial analysis to relate the existence of flat bands to vertical faces of the generic Newton polytope. A key insight is that vertical faces force potential-independence and lead to a 0-support component in a fundamental domain, which is then ruled out generically via resultant arguments and induction on the vertex count. Consequently, for generic labeling, $D(z,\lambda)$ has no linear factor in $\lambda$, establishing the discrete analogue of a broad conjecture on the rarity of flat bands in periodic operators and clarifying when flat bands can occur (only in degenerate 0-support scenarios). The results unify combinatorics, algebraic geometry, and spectral theory to address a longstanding question in the spectral theory of periodic graphs and their Bloch varieties.

Abstract

As a corollary of our main results, we prove that for any connected $\mathbb{Z}^d$-periodic graph, when edge weights and potentials are treated as variables, the corresponding periodic graph operators generically (i.e., outside a proper algebraic subset of the variable space) do not have flat bands.

Figures (1)

• Figure 1: The ${\mathbb Z}^2$-periodic Lieb lattice.

Theorems & Definitions (46)

• Theorem 1.1
• Example 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5