Green parafermions as emergent flat-band excitations in condensed matter

Abstract

Green parafermions, originally introduced by Green and extended by Greenberg and Messiah through trilinear and relative trilinear commutation relations beyond Bose-Fermi statistics, are generally regarded as mathematical curiosities without physical realization. We show that these paraparticles can in fact emerge as composite excitations in a broad class of condensed-matter systems undergoing spontaneous symmetry breaking with type-B Goldstone modes. The key ingredient is the introduction of auxiliary Majorana fermions defined on emergent unit cells produced by partial translational-symmetry breaking. When the auxiliary Majoranas are treated as physical degrees of freedom, the resulting Green parafermion states (up to a projection operator) correspond to flat-band excitations, whose creation and annihilation operators satisfy the trilinear algebra. When they are regarded as fictitious, the same construction explains the appearance of exponentially many degenerate ground states and reveals a surprising correspondence between Green parafermions and self-similar geometric objects, such as the golden spiral. Explicit realizations are demonstrated for the ferromagnetic spin-1 biquadratic model and the ferromagnetic $\rm {SU}(2)$ flat-band Tasaki model, showing that condensed-matter systems with type-B Goldstone modes provide a natural setting for Green parafermions as emergent, possibly observable quasiparticles.