The nonlinear Dirac equation in Bose-Einstein condensates: Vortex solutions and spectra in a weak harmonic trap

TL;DR

This paper explores vortex solutions of the nonlinear Dirac equation in Bose-Einstein condensates within a honeycomb lattice, revealing various topological structures and their spectra in a weak harmonic trap.

Contribution

It provides a comprehensive derivation and classification of vortex solutions, including skyrmions and half-quantum vortices, and analyzes their spectral properties in a trapped setting.

Findings

Identified multiple vortex types including skyrmions and half-quantum vortices.

Derived explicit solutions using asymptotic series, algebraic forms, and numerical methods.

Mapped the spectral transition from vortex states to free particles.

Abstract

We analyze the vortex solution space of the $(2 +1)$-dimensional nonlinear Dirac equation for bosons in a honeycomb optical lattice at length scales much larger than the lattice spacing. Dirac point relativistic covariance combined with s-wave scattering for bosons leads to a large number of vortex solutions characterized by different functional forms for the internal spin and overall phase of the order parameter. We present a detailed derivation of these solutions which include skyrmions, half-quantum vortices, Mermin-Ho and Anderson-Toulouse vortices for vortex winding $\ell = 1$. For $\ell \ge 2$ we obtain topological as well as non-topological solutions defined by the asymptotic radial dependence. For arbitrary values of $\ell$ the non-topological solutions are bright ring-vortices which explicitly demonstrate the confining effects of the Dirac operator. We arrive at solutions through an asymptotic Bessel series, algebraic closed-forms, and using standard numerical shooting methods. By including a harmonic potential to simulate a finite trap we compute the discrete spectra associated with radially quantized modes. We demonstrate the continuous spectral mapping between the vortex and free particle limits for all of our solutions.