Large-momentum tail of one-dimensional Fermi gases with spin-orbit coupling

We study the contacts, large-momentum tail, and universal relations for an ultracold one-dimensional (1D) two-component Fermi gas with spin-orbit coupling (SOC). Different from previous studies, we find that the $q^{-8}$ tail in the spin-mixing (off-diagonal) terms of the momentum distribution matrix is dependent on the two SOC parameters in the laboratory frame for 1D systems, where $q$ is the relative momentum. This tail can be observed through time-of-flight measurement as a direct manifestation of the SOC effects on the many-body level. Besides the traditional $s$-wave scattering length, we find that two new contacts need to be introduced due to the SOC. Consequently, two new adiabatic energy relations with respect to the two SOC parameters are obtained. Further, we derive the pressure relation and virial theorem at short distance for this system. To find how the SOC modifies the large-momentun behavior, we take the SOC parameters as perturbations, since the strength of the SOC should be much smaller than the corresponding strength scale of the interatomic interactions. In addition, by using the operator product expansion method, we derive the asymptotic behavior of the large-momentum distribution matrix up to the $q^{-8}$ order, and find that the diagonal terms of the distribution matrix include the contact of traditional $s$-wave scattering length as the leading term and the SOC modified terms beyond the leading term, the off-diagonal term is beyond subleading term and is corrected by the SOC parameters. We also find that the momentum distribution matrix shows spin-dependent and anisotropic features. Furthermore, we calculate the momentum distribution matrix in the laboratory frame for the experimental implication. This work paves the way for exploring the profound properties of many-body quantum systems with SOC in one dimension.