Solitons in lattice field theories via tight-binding supersymmetry

Abstract

Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems, non-perturbative solutions of various large-N field theories, and closely related solitonic solutions to the Bogoliubov-de Gennes equations in the theory of superconductivity. These solutions rely on the inverse scattering method, which reduces these seemingly unrelated problems to identifying reflectionless potentials of an auxiliary one-dimensional quantum scattering problem. There are several ways of constructing these potentials, one of which is quantum mechanical supersymmetry (SUSY). In this paper, motivated by recent experimental platforms, we generalize this framework to develop a theory of solitons in lattice versions of interacting field theories. Our analysis hinges on the use of trace identities, which are relations connecting the potential of an equation of motion to the scattering data. For a discrete Schrödinger operator, such trace identities are well known, and we derive a new set of identities for the discrete Dirac operator. We then use these in a lattice Gross-Neveu and chiral Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond to reflectionless potentials associated with the discrete scattering problem. These models are of significance as they are equivalent to a mean-field theory of a lattice superconductor. To explicitly construct these solitons, we generalize supersymmetric quantum mechanics to tight-binding models. We show that a matrix transformation exists that maps a tight-binding model to an isospectral one which shares the same structure and scattering properties. The corresponding soliton solutions have both modulated hopping and onsite potential; we explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.

arXiv comment42 pages, 4 figures

References in corpus (25)

• Supersymmetry and Quantum Mechanics
• Formation and Propagation of Matter Wave Soliton Trains
• Formation of a Matter-Wave Bright Soliton
• Dark Solitons in Bose-Einstein Condensates
• Solitons in nonlinear lattices
• Quantum Time Crystals
• The Fermi-Pasta-Ulam problem: 50 years of progress
• Solitons and Rabi Oscillations in a Time-Dependent BCS Pairing Problem
• Inhomogeneous Condensates in the Thermodynamics of the Chiral NJL_2 model
• Heavy Solitons in a Fermionic Superfluid
• Algebraic Aspects of Bethe-Ansatz
• Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems
• A Twisted Kink Crystal in the Chiral Gross-Neveu model
• Topological solitons in three-band superconductors with broken time reversal symmetry
• The Bogoliubov/de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry
• Self-consistent multiple complex-kink solutions in Bogoliubov-de Gennes and chiral Gross-Neveu systems
• Normal and anomalous solitons in the theory of dynamical Cooper pairing
• Phase solitons in multi-band superconductors with and without time-reversal symmetry
• Fermionic solutions of chiral Gross-Neveu and Bogoliubov-de Gennes systems in nonlinear Schrödinger hierarchy
• Moving solitons in a one-dimensional fermionic superfluid
• Twisted kinks, Dirac transparent systems and Darboux transformations
• Transparent Dirac potentials in one dimension: the time-dependent case
• Non-perturbative Quantum Dynamics of the Order Parameter in the Pairing Model
• On reflectionless nature of self-consistent multi-soliton solutions in Bogoliubov-de Gennes and chiral Gross-Neveu models
• Phase transition into Instanton Crystal