Spin-charge separation for paired Dirac fermions in (1 + 1) dimensions

We study Dirac fermions at finite density coupled to a complex pairing field assumed to obey scalar field theory with quartic self-repulsion. The bulk of our work develops the mathematics that elucidates the propagation of fermionic excitations in such systems as independent spin (boosts) and charge (fermion number) degrees of freedom. A necessary ingredient is the presence of broken U(1) symmetry in the pairing field and decoupling of its density and phase. In the fermion sector, these elements give rise to an emergent spin-dependent gauge coupling which binds in-vacuum spin and charge into elementary fermions, while driving proliferation of unbound spin and charge for finite condensation in the pairing field. Notably, the onset of spin-charge separation is signaled by \( \mathcal{PT} \)-symmetry breaking and decoupling of spin components under Lorentz transformations. Our investigation concludes with two theorems that identify generic features of spin-charge separation in such systems.