  We investigate the superfluidity of a two-component Fermi gas with
spin-orbital-angular-momentum coupling (SOAMC). Due to the intricate interplay
of SOAMC, two-photon detuning and atom-atom interaction, a family of vortex
ground states emerge in a broad parameter regime of the phase diagram, in
contrast to the usual case where an external rotation or magnetic field is
generally required. More strikingly, an unprecedented vortex state, which
breaks the continuous rotational symmetry to a discrete one spontaneously, is
predicted to occur. The underlying physics are elucidated and verified by
numerical simulations. The unique density distributions of the predicted vortex
states enable a direct observation in experiment.
