Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates

Abstract

We present relativistic linear stability equations (RLSE) for quasi-relativistic cold atoms in a honeycomb optical lattice. These equations are derived from first principles and provide a method for computing stabilities of arbitrary localized solutions of the nonlinear Dirac equation (NLDE), a relativistic generalization of the nonlinear Schrödinger equation. We present a variety of such localized solutions: skyrmions, solitons, vortices, and half-quantum vortices, and study their stabilities via the RLSE. When applied to a uniform background, our calculations reveal an experimentally observable effect in the form of Cherenkov radiation. Remarkably, the Berry phase from the bipartite structure of the honeycomb lattice induces a boson-fermion transmutation in the quasi-particle operator statistics.

6 pages, 3 figures

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• Relativistic quantum effects of Dirac particles simulated by ultracold atoms
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• Dynamics of Dirac solitons in networks
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• The nonlinear Dirac equation in Bose-Einstein condensates: II. Relativistic soliton stability analysis
• Optical analogue of spontaneous symmetry breaking induced by tachyon condensation in amplifying plasmonic arrays
• Nonlinear conformal-degree preserving Dirac equations
• Honeycomb supersolid -- Dirac points and shear-instability induced crystal transitions
• Speed-of-light pulses in a nonlinear Weyl equation
• Bose-Einstein condensate of Dirac magnons: Pumping and collective modes
• Theory of hydrodynamic phenomena in optical mesh lattices