Universal Relations of Ultracold Fermi Gases with Arbitrary Spin-Orbit Coupling

We derive the universal relations for an ultracold two-component Fermi gas with spin-orbit coupling (SOC) $\sum_{\alpha,\beta=x,y,z}\lambda_{\alpha\beta}\sigma_{\alpha}p_{\beta}$, where $p_{x,y,z}$ and $\sigma_{x,y,z}$ are the single-atom momentum and Pauli operators for pseudo spin, respectively, and the SOC intensity $\lambda_{\alpha\beta}$ could take arbitrary value. We consider the system with an s-wave short-range interspecies interaction, and ignore the SOC-induced modification for the value of the scattering length. Using the first-quantized approach developed by S. Tan (Phys. Rev. Lett. \textbf{107}, 145302 (2011)), we obtain the short-range and high-momentum expansions for the one-body real-space correlation function and momentum distribution function, respectively. For our system these functions are $2\times2$ matrix in the pseudo-spin basis. We find that the leading-order ($1/k^{4}$) behavior of the diagonal elements of the momentum distribution function (i.e., $n_{\uparrow\uparrow}({\bf k})$ and $n_{\downarrow\downarrow}({\bf k})$) are not modified by the SOC. However, the SOC can significantly modify the behavior of the \textit{non-diagonal elements} of the momentum distribution function, i.e., $n_{\uparrow\downarrow}({\bf k})$ and $n_{\downarrow\uparrow}({\bf k})$, in the large-$k$ limit. In the absence of the SOC, the leading order of these elements is ${\cal O}(1/k^{6})$. When SOC appears, it can induce a term on the order of $1/k^{5}$. We further derive the adiabatic relation and the energy functional. Our results show the SOC can induce a new term in the energy functional, which simply describe the contribution from the SOC to the total energy. The form of the adiabatic relation for our system is not modified by the SOC.