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

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 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.

Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

“…They provide the identification of the nontrivial integrals of motion as fermionic and bosonic generators in a way different from that described here. Particularly, the treatment of P n;a as odd supercharges is possible, see [24,37,46,47,49,58]. Supersymmetric structures for alternative choices of Γ can be computed by employing the product relations of the intertwining generators and Lax operators collected in Appendix.…”

“…As both families of quantum systems are characterized by nontrivial, higher derivative integrals of motion, one could expect that supersymmetric extensions of them should possess some peculiar properties. This is indeed the case [16,17,18,19,20,21], and exotic supersymmetric structures of reflectionless and finite-gap systems found recently some interesting physical applications [22,23,24,25,26].…”

“…1 was investigated and applied for the description of the kink and kink-anti-kink solutions of the Gross-Neveu model [46,24]. One can expect that the self-isospectral pair of reflectionless Pöschl-Teller systems with n > 1 bound states should also be described by some not studied yet exotic nonlinear supersymmetric structure.…”

“…Such a kind of supersymmetry of the pairs of reflectionless systems was not investigated yet in the literature, but, instead, supersymmetry of the pairs (H + = H j , H − = H j+l ), l ≥ 1, belonging to the same Darboux chain (2.12) is usually considered. In particular, the pairs of reflectionless Pöschl-Teller systems, see below, appear in the context of shape-invariance [43,44,7], they also emerge in the infinite-period limit of finite-gap periodic crystal structures [22,24]. Supersymmetry of reflectionless Pöschl-Teller pairs (H j , H j+l ) was studied recently from the perspective of AdS/CFT holography and Aharonov-Bohm effect [45].…”

“…There are other possibilities to choose Γ, which are based on reflection operators and classify the nontrivial integrals of the extended system in a way different from that prescribed by the choice Γ = σ 3 . The alternative choices for Γ find some interesting physical applications, see [22,24,46,47], and we return to the discussion of this point in the last Section.…”