Self-isospectrality, special supersymmetry, and their effect on the band structure

We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lame equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parity-odd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking.