Splitting of the Fermi point of strongly interacting electrons in one dimension: A nonlinear effect of spin-charge separation

A system of one-dimensional electrons interacting via a short-range potential described by Hubbard model is considered in the regime strong coupling using Bethe ansatz approach. We study its momentum distribution function at zero temperature and find one additional singularity $3{k}_{\mathrm{F}}$ point. identify that second same Luttinger liquid type as low-energy ${k}_{\mathrm{F}}$. By calculating spectral simultaneously, we show formed charge modes only, unlike known around ${k}_{\mathrm{F}}$ consisting both spin modes. This result reveals ability spin-charge separation effect to split Fermi point free into two, demonstrating robustness beyond limit where it was originally found.