Theoretical progress and material studies of heavy fermion superconductors

Heavy fermion superconductors belong to a special class of strongly correlated systems and unconventional superconductors. The emergence of superconductivity in these materials is closely associated with the presence of quantum critical fluctuations. Heavy fermion superconductors of different structures often exhibit distinct competing orders and superconducting phase diagrams, implying sensitive dependence of their electronic structures and pairing mechanism on the crystal symmetry. Here we give a brief introduction on recent theoretical and experimental progress in several different material families. We develop a new phenomenological framework of superconductivity combining the Eliashberg theory, a phenomenological form of quantum critical fluctuations, and strongly correlated band structure calculations for real materials. Our theory provides a unified way for systematic understanding of various heavy fermion superconductors.