Irreducibility of the Bloch variety for elliptic periodic operators

Establish the irreducibility of the Bloch variety of elliptic periodic operators, in order to confirm that realistic fully periodic material models cannot possess a global block-diagonal band structure decomposition across the Brillouin zone.

Background

The paper analyzes when the Brillouin Complex Deformation (BCD) method fails, highlighting artificial examples with global block-diagonal structures where band crossings abound and BCD is unreliable. The authors argue that realistic fully periodic (bulk) materials are not expected to exhibit such global block-diagonal decompositions, linking this expectation to a broader conjecture in spectral theory.

Specifically, they reference a conjecture about the irreducibility of the Bloch variety for elliptic periodic operators. If the Bloch variety were reducible, it could admit global decompositions that would make such block structures natural; thus, proving irreducibility supports the expectation that global block-diagonality does not occur in realistic systems.

References

Realistic models of materials (with full periodicity, i.e. excluding surfaces which are 3D models with 2D periodicity) are not expected to have this structure globally; indeed, this would contradict the conjectured irreducibility of the Bloch variety of elliptic periodic operators .

— Numerical methods for the computation of densities of states of periodic operators  (2603.29457 - Lallinec et al., 31 Mar 2026) in Section “Validity of the BCD”, Subsection “Discussion”