Exotic Supersymmetry of the Kink-Antikink Crystal, and the Infinite Period Limit

Some time ago, Thies showed that the Gross-Neveu model with a bare mass term possesses a kink-antikink crystalline phase. Corresponding self-consistent solutions, known earlier in polymer physics, are described by a self-isospectral pair of one-gap periodic Lamé potentials with a Darboux displacement depending on the bare mass. We study an unusual supersymmetry of such a second-order Lamé system, and show that the associated first-order Bogoliubov-de Gennes Hamiltonian possesses its own nonlinear supersymmetry. The Witten index is ascertained to be zero for both of the related exotic supersymmetric structures, each of which admits several alternatives for the choice of a grading operator. A restoration of the discrete chiral symmetry at zero value of the bare mass, when the kink-antikink crystalline condensate transforms into the kink crystal, is shown to be accompanied by structural changes in both of the supersymmetries. We find that the infinite period limit may or may not change the index. We also explain the origin of the Darboux-dressing phenomenon recently observed in a nonperiodic self-isospectral one-gap Pöschl-Teller system, which describes the Dashen, Hasslacher, and Neveu kink-antikink baryons. © 2011 American Physical Society.