Abstract
Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), 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 lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to 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 had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities 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, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.
Highlights
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
Fermi and co-workers were interested in the question of thermalization and sought to observe it in a discrete chain of non-linear oscillators. They numerically observed a recurrent, almost periodic behavior instead, which became known as the Fermi-Pasta-Ulam-Tsingou paradox [15]. This paradox motivated the work by Kruskal and Zabusky ten years later, who showed that the FPUT problem maps onto the Korteweg-de Vries (KdV) equation in the continuum limit [2]
Since this formulation can be rather bulky, we introduce an alternate formalism based on a generalization of supersymmetric quantum mechanics to tight binding models, and use this method to construct the kink soliton and kink-antikink solutions described in the previous section, without resorting to using the Gelfand-Levitan equation
Summary
We start with a review of soliton solutions in continuum systems. The simplest model which exhibits such solutions is the Korteweg-de Vries equation, which was first used to describe shallow water waves [1]. The mean field theory of a chiral version of the Gross-Neveu model, known as the Nambu-Jona-Lasinio model, can be shown to be equivalent to BCS superconductivity in 1D [13, 30] Both the Gross-Neveu and the chiral Gross-Neveu model have nontrivial saddle point solutions which correspond to solitons. It turns out that the construction of soliton solutions for the KdV equation and the Gross-Neveu field theories are nearly identical. This is because both the KdV equation as well as the equations of motion for the Gross-Neveu model can be mapped onto a 1D quantum scattering problem.
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