Prof. Renjun Xu
Many years ago, being a Ph.D. student I asked the same questions to some famous fellow Profs. One answered to me that in order to have an Ab initio simulation code for superconductivity, the pairing (interaction mechanism) should be well established. Of course, that is the main issue in many superconductors, mostly unconventional ones.
Another complication is the impurities (magnetic in BCS & nonmagnetic in unconventional with nodes/quasi-nodes) again the symmetry of the pairing araises, this time with the disorder included (magnetic or non-magnetic).
Finally, there is one more issue, self-consistency in several quantities, such as energy, potential, and even dimensionality. A really difficult subject. To make harder the question, add please that in some unconventional superconductors there are incommensurable magnetic phases, and to finally get it harder, there can be several thermodynamic phases with different types of pairing (singlet, triplet) or different phases (superconducting, antiferromagnetic) in the same material.
Thank you for the interesting question.
Best Regards.
Dear Professor Xu,
Currently there is no unified theory, which could predict SC in any materials. So there are no rules to check if the new matter is a conventional or unconventional superconductor. Reasonably, a correct unified theory must introduce quantitative parameters for all materials and, thus, eliminate qualitative differences between conventional/unconventional superconductors.
Thus the question is very actual.