Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit

TL;DR

This paper explores the exotic supersymmetry properties of kink-antikink crystals in the Gross-Neveu model, revealing novel nonlinear supersymmetries, their behavior under the infinite period limit, and connections to Darboux dressing phenomena.

Contribution

It uncovers the nonlinear supersymmetry of the Lame system associated with kink-antikink crystals and analyzes the effects of the infinite period limit on supersymmetry and the Witten index.

Findings

The associated Bogoliubov-de Gennes Hamiltonian exhibits nonlinear supersymmetry.

The Witten index is zero for the exotic supersymmetric structures.

The infinite period limit can preserve or alter the supersymmetry index.

Abstract

Some time ago, Thies et al. 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 Lame potentials with a Darboux displacement depending on the bare mass. We study an unusual supersymmetry of such a second order Lame system, and show that the associated first order Bogoliubov-de Gennes Hamiltonian possesses the 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 non-periodic self-isospectral one-gap Poschl-Teller system, which describes kink-antikink baryons of Dashen, Hasslacher and Neveu.