The Nonlinear Dirac Equation in Bose-Einstein Condensates: Foundation and Symmetries

TL;DR

This paper derives a nonlinear Dirac equation for Bose-Einstein condensates in honeycomb lattices, highlighting how it differs from particle physics models by breaking Poincaré symmetry, and discusses the associated symmetries.

Contribution

It provides a rigorous derivation of the nonlinear Dirac equation for BECs and analyzes the symmetry properties specific to this physical system.

Findings

Derivation of a nonlinear Dirac equation from first principles for BECs in honeycomb lattices.

Identification of broken and preserved symmetries in the nonlinear Dirac model.

Clarification of differences between BEC nonlinear Dirac equations and those in particle physics.

Abstract

We show that Bose-Einstein condensates in a honeycomb optical lattice are described by a nonlinear Dirac equation in the long wavelength, mean field limit. Unlike nonlinear Dirac equations posited by particle theorists, which are designed to preserve the principle of relativity, i.e., Poincar\'e covariance, the nonlinear Dirac equation for Bose-Einstein condensates breaks this symmetry. We present a rigorous derivation of the nonlinear Dirac equation from first principles. We provide a thorough discussion of all symmetries broken and maintained.