Fractional Vortices in $J = 2$ Condensates

We consider the possible ground-states and topologically stable line defects in BCS and BEC condensates with total spin $J=2$, including spinor BECs, as well BCS condensates with total angular momentum $J=2$. For cold Fermi gases it may be possible to realize $^{1}D_{2}$ or $^{3}P_{2}$ condensates of BCS pairs described by a symmetric and traceless matrix, $A_{\mu\nu}$, for the $2J+1=5$ complex amplitudes that transform as a rank 2 tensor under joint spin and orbital rotations. Condensates with $J=2$ have a rich phase diagram. We discuss the residual symmetry and fundamental group of $J=2$ condensates exhibiting \emph{complex, bi-axial} order, $A_{\mu\nu}=\Delta\,e^{i\varphi} \left[u_{\mu}u_{\nu}+\epsilon v_{\mu} v_{\nu}+\epsilon^2 w_{\mu} w_{\nu}\right]$, where $\epsilon=e^{i\,2\pi/3}$ and $u,v,w$ are an othogonal triad. This remarkable phase has tetrahedral point symmetry and is described by a non-abelian fundamental group $\pi_{1}(G/H)$. We classify the topologically stable line defects and show that conventional $U(1)$ phase vortices can dissociate into \emph{fractional} vortices with $2 \pi/3$ phase winding combined with tetrahedral rotations, indexed by the conjugacy classes of the non-abelian isotropy subgroup $H$, and consider associated fermionic bound states.