The nonlinear Dirac equation in Bose-Einstein condensates: Superfluid fluctuations and emergent theories from relativistic linear stability equations

TL;DR

This paper develops a comprehensive theoretical framework for analyzing the stability and excitations of Bose-Einstein condensates at Dirac points, revealing connections to various fundamental physics equations and phenomena.

Contribution

It derives the relativistic linear stability equations for BECs at Dirac points using two methods and explores their implications for vortex stability and emergent physical theories.

Findings

Vortex solutions have lifetimes of approximately 4 seconds.

RLSE reduce to known equations like Andreev, Majorana, and Dirac-Bogoliubov-de Gennes under specific limits.

Tuning parameters reveals regimes of BCS superconductivity and fundamental wave equations.

Abstract

We present the theoretical and mathematical foundations of stability analysis for a Bose-Einstein condensate (BEC) at Dirac points of a honeycomb optical lattice. The combination of s-wave scattering for bosons and lattice interaction places constraints on the mean-field description, and hence on vortex configurations in the Bloch-envelope function near the Dirac point. A full derivation of the relativistic linear stability equations (RLSE) is presented by two independent methods to ensure veracity of our results. Solutions of the RLSE are used to compute fluctuations and lifetimes of vortex solutions of the nonlinear Dirac equation, which include Mermin-Ho and Anderson-Toulouse skyrmions, with lifetime $\approx 4$ seconds. Beyond vortex stabilities the RLSE provide insight into the character of collective superfluid excitations, which we find to encode several established theories of physics. In particular, the RLSE reduce to the Andreev equations, in the nonrelativistic and semiclassical limits, the Majorana equation, inside vortex cores, and the Dirac-Bogoliubov-de Gennes equations, when nearest-neighbor interactions are included. Furthermore, by tuning a mass gap, relative strengths of various spinor couplings, for the small and large quasiparticle momentum regimes, we obtain weak-strong Bardeen-Cooper-Schrieffer superconductivity, as well as fundamental wave equations such as Schr\"odinger, Dirac, Klein-Gordon, and Bogoliubov-de Gennes equations. Our results apply equally to a strongly spin-orbit coupled BEC in which the Laplacian contribution can be neglected.