Superconducting classes of heavy-fermion materials

Stimulated by recent remarkable findings which lead to a common feature of all four known heavy-fermion superconductors ${\mathrm{UPt}}_3$, (U,Th)${\mathrm{Be}}_{13}$, ${\mathrm{URu}}_2$ ${\mathrm{Si}}_2$, and ${\mathrm{CeCu}}_2$ ${\mathrm{Si}}_2$, namely, a close proximity between the magnetism and unconventional superconducting pairing state we present a general framework on how to identify the superconducting classes. Taking into account the existence of the spin-density wave (SDW) above $T_c$ in ${\mathrm{UPt}}_3$ and ${\mathrm{URu}}_2$ ${\mathrm{Si}}_2$ and below $T_c$ in (U,Th)${\mathrm{Be}}_{13}$ and ${\mathrm{CeCu}}_2$ ${\mathrm{Si}}_2$, we enumerate group theoretically the pairing states compatible with given space-group symmetry according to the Landau subordination scheme for the second-order phase transition. This enables us to effectively narrow down the possible states. These enumerated states are quite general, independent of particular models or pairing interaction forms. Then we give some physical arguments on the relative stability among the allowed states. Maximally utilizing experimental and theoretical data available at present, we attempt to decrease the number of the possible states and obtain the most plausible candidates for each material. To confirm the emerging universal feature that the ground state in heavy-fermion superconductors is the coexisting phase with the magnetism, further experiments are proposed, based on our results.