Theoretical framework demonstrates an exact polynomial GCD certificate for flat bands in periodic tight-binding lattices, highlighting its utility for hopping parameter engineering.
Key Points
Establish an exact polynomial greatest common divisor (GCD) certificate to identify dispersionless flat bands and their algebraic multiplicities in finite-range periodic tight-binding systems.
Expressed the characteristic polynomial of the Bloch Hamiltonian as a Laurent polynomial in lattice momentum variables.
Extracted the monic greatest common divisor over all energy-dependent polynomial coefficients.
Evaluated the algebraic certificate on kagome, dice, and octahedron-chain lattices, including weighted variants.
Demonstrated that the roots of the monic GCD correspond exactly to flat-band energies, with root multiplicities reflecting Brillouin-zone algebraic multiplicities.
Confirmed that the certificate remains invariant under unit cell redefining and Bloch-gauge transformations.
Established the framework's effectiveness as a symbolic computation tool for analytical hopping parameter engineering.