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

TL;DR

This paper introduces relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates, enabling analysis of localized solutions and revealing observable effects like Cherenkov radiation and boson-fermion transmutation.

Contribution

The paper derives RLSE from first principles for the NLDE in honeycomb lattices and applies them to various localized solutions, highlighting novel stability analysis methods.

Findings

Identification of stable skyrmions, solitons, vortices, and half-quantum vortices.

Prediction of Cherenkov radiation in uniform backgrounds.

Discovery of Berry phase-induced boson-fermion transmutation.

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 Schr\"odinger 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.