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

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.

37 pages, 15 figures

References in corpus (9)

• Topological Insulators with Inversion Symmetry
• Spin-orbit coupled Bose-Einstein condensates
• Making Massless Dirac Fermions from Patterned Two-Dimensional Electron Gases
• Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene
• Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity
• Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates
• Non-Abelian vortices in Chern-Simons theories and their induced effective theory
• Honeycomb optical lattices with harmonic confinement
• BPS Vortices in Non-relativistic M2-brane Chern-Simons-matter Theory

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• ${\mathbb{Z}}_2$ topological insulator analog for vortices in an interacting bosonic quantum fluid
• Stability of solitary waves and vortices in a 2D nonlinear Dirac model
• Ring Dirac Solitons in Nonlinear Topological Lattices
• Spinor solitons and their $\mathcal{P}\mathcal{T}$-symmetric offspring
• Solitary waves of a PT-symmetric Nonlinear Dirac equation
• Solitary waves in the Nonlinear Dirac Equation
• Dirac Solitons and Topological Edge States in the $\beta$-Fermi-Pasta-Ulam-Tsingou dimer lattice
• Speed-of-light pulses in a nonlinear Weyl equation
• Bose-Einstein condensate of Dirac magnons: Pumping and collective modes