Twisted kinks, Dirac transparent systems and Darboux transformations

Abstract

Darboux transformations are employed in construction and analysis of Dirac Hamiltonians with pseudoscalar potentials. By this method, we build a four parameter class of reflectionless systems. Their potentials correspond to composition of complex kinks, also known as twisted kinks, that play an important role in the $1+1$ Gross-Neveu and Nambu-Jona-Lasinio field theories. The twisted kinks turn out to be multi-solitonic solutions of the integrable AKNS hierarchy. Consequently, all the spectral properties of the Dirac reflectionless systems are reflected in a non-trivial conserved quantity, which can be expressed in a simple way in terms of Darboux transformations. We show that the four parameter pseudoscalar systems reduce to well-known models for specific choices of the parameters. An associated class of transparent non-relativistic models described by matrix Schrödinger Hamiltonian is studied and the rich algebraic structure of their integrals of motion is discussed.

11 pages, 2 figures, published version

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• Soliton defects in one-gap periodic system and exotic supersymmetry
• Minimal Realizations of Supersymmetry for Matrix Hamiltonians
• Flat-band engineering of quasi-one-dimensional systems via supersymmetric transformations
• Reflectionless pseudospin-1 Dirac systems via Darboux transformation and flat band solutions
• Two-dimensional Dirac fermion in presence of an asymmetric vector potential
• Lorentzian quantum wells in graphene: the role of shape invariance in zero-energy states trapping
• SUSY design of smooth quantum rings in graphene
• Solitons in lattice field theories via tight-binding supersymmetry
• Coupled system of Dirac fermions with different Fermi velocities via composites of SUSY operators
• The soliton nature of the super-Klein tunneling effect
• Polynomial supersymmetry for matrix Hamiltonians: proofs