Effects of spin-orbit interaction in the Hubbard chain with attractive interaction: Application to confined ultracold fermions

Motivated by recent experiments on ultracold fermions we study the effect of the spin-orbit interaction in an exactly solvable one-dimensional model with strong local attractive correlations between the fermions. We show that the asymptotes of correlation functions, calculated in the framework of the conformal field theory and finite size corrections of the Bethe ansatz exact solution, are strongly affected by the spin-orbit coupling. In the mixed phase, the correlation functions consist of terms that are the product of a power law of the distance and an oscillating function of the distance. We obtain the critical exponents for superfluidity and density waves. The leading exponents decrease as a function of spin-orbit coupling. In particular, the exponent for superfluidity is the smallest one signaling an instability to a phase where a weak interchain coupling can produce superfluidity of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type. The spatial oscillation of the order parameter is also modified by the spin-orbit parameter, i.e., its modulation is not just given by the spin imbalance.

Figure 1

Critical exponents for the asymptotes of correlation functions for Cooper pairs (${\theta }_{\text{CP}}$, solid curve) and density waves for an attractive Hubbard interaction $U=|u/t|=8$ at quarter-filling as a function of the SOI phase $\varphi$. The four cases presented, $H=5.0$, 5.7, 7.2, and 8.4, correspond to the mixed phase, i.e., coexisting pairs and unpaired particles. The exponent ${\theta }_{\text{DW}00}$ (dashed curve) refers to $D_u=D_p=0$, ${\theta }_{\text{DW}01}$ (dotted curve) to $D_u=0$ and $D_p=1$, and ${\theta }_{\text{DW}11}$ (dashed-dotted curve) to $D_u=D_p=1$.