Contact Tensor in a $p$-Wave Fermi Gas with Anisotropic Feshbach Resonances

Recent theoretical and experimental investigations have revealed that a Fermi gas with a p-wave Feshbach resonance has universal relations between the system's high-momentum behavior and thermodynamics. A new feature introduced by the p-wave interaction is anisotropy in the Feshbach resonances; three degenerate $p$-wave resonances split according to the magnetic quantum number of the closed-channel molecules $|m|$ due to the magnetic dipole-dipole interaction. Here, we investigate the consequences of the anisotropy. We show that the momentum distribution has a high-momentum asymptote $n_k \sim k^{-2} \sum_{m,m'=-1}^1 C_{m,m'}Y_1^{m\ast}(\hat{k}) Y_1^{m'}(\hat{k})$, in which we introduce the $p$-wave contact tensor $C_{m,m'}$. In contrast to the previous studies, it has nine components [1,2]. We identify them as the number, angular momentum, and nematicity of the closed-channel molecules. We also discuss two examples, the anisotropic $p$-wave superfluid and a gas confined in a cigar-shaped trap, which exhibit a nematicity component in the $p$-wave contact tensor. [1] S. M. Yoshida, and M. Ueda, Phys.~Rev.~Lett.~\textbf{115}, 135303 (2015). [2] Z. Yu, J. H. Thywissen, and S. Zhang, Phys.~Rev.~Lett.~\textbf{115}, 135304 (2015).