Critical Point Conjecture for Bloch Varieties of Discrete Periodic Operators

Establish that for generic parameters V and E defining a discrete periodic operator H on a Zd-periodic graph Γ with dispersion polynomial D(z,λ), every solution (z,λ0) of the critical point equations D(z,λ)=∂D/∂z1(z,λ)=⋯=∂D/∂zd(z,λ)=0 is smooth and isolated on the complexified Bloch variety; equivalently, show that the Jacobian matrix of these equations is invertible at (z,λ0).

Background

The Bloch variety is the complexification of the dispersion relation obtained from the dispersion polynomial D(z,λ) via Floquet theory. Critical points are solutions to the system D(z,λ)=0 together with vanishing ∂D/∂zi for all coordinates.

The paper recalls the Critical Point Conjecture from prior works and proves that it implies the Spectral Edges Conjecture. The authors verify the conjecture for specific families of periodic graphs, but the general validity of the conjecture for arbitrary graphs and parameters remains open.