Quantum valence criticality in heavy fermions on periodic and aperiodic crystals

Progress of theories and experiments of the quantum valence criticality is overviewed focusing on recent discoveries of direct evidence of the quantum critical point of the valence transition in the periodic crystal α-YbAl_1-xFe_xB₄ at x = 0.014 and the quasicrystal Yb₁₅(Al_1-xGa_x)₃₄(Au_1-yCu_y)₅₁ at x = y = 0. The common quantum criticality as well as T/B scaling behavior in the magnetization observed in the periodic crystal β-YbAlB₄ and the quasicrystal Yb₁₅Al₃₄Au₅₁ is shown to be explained by the theory of critical Yb-valence fluctuations starting from the Hamiltonians for describing the low-energy electronic states in both systems. (author)