Unconventional color superfluidity in ultra-cold fermions: Quintuplet pairing, quintuple point and pentacriticality

We describe the emergence of color superfluidity in ultra-cold fermions induced by color-orbit and color-flip fields that transform a conventional singlet-pairing s-wave system into an unconventional non-s-wave superfluid with quintuplet pairing. We show that the tuning of interactions, color-orbit and color-flip fields transforms a momentum-independent scalar order parameter into an explicitly momentum-dependent tensor order parameter. We classify all unconventional superfluid phases in terms of the {\it loci} of zeros of their quasi-particle excitation spectrum in momentum space and we identify several Lifshitz-type topological transitions. Furthermore, when boundaries between phases are crossed, non-analyticities in the compressibility arise. We find a quintuple point, which is also pentacritical, where four gapless superfluid phases converge into a fully gapped superfluid phase.