A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schrödinger operators

Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schrödinger operator is Fermi isospectral to the zero potential. This provides a negative answer to a question posed by the third author in [29] concerning the rigidity of Fermi isospectrality in dimension two. This example also disproves a conjecture of Gieseker, Knörrer, and Trubowitz in the 1990 s stating that for any nontrivial real-valued periodic potential in dimension two, the Fermi variety is irreducible at all energy levels.