Thank you for the comment. I would like to have a flat band which covers just a part of momentum space, not the full space (see the last paragraph of the modified question). I will read the paper you mentioned.

By Weyl crossings I meant a point in the Brillouin zone, where two bands touch each other. Near this point the energy dispersion is linear and the system can be described by the Weyl equation.

At T=0 the system is a chiral superfluid for any strength of the interatomic attraction. The BEC and BCS regimes are however topologically different and are separated by a quantum phase transition.

In general, the order of the phase transition tells us how bad the non-analyticity of the thermodynamics potential is at the phase transition point. The third order phase transition has a discontinuity in the third derivative of the thermodynamic potential, while the first and the second derivatives are continuous.