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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 quantum critical point of valence transitions in periodic crystal $α$-YbAl$_{1-x}$Fe$_x$B$_4$ at $x=0.014$ and quasicrystal Yb$_{15}$(Al$_{1-x} $Ga$_{x}$)$_{34}$(Au$_{1-y}$Cu$_{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 $β$-YbAlB$_4$ and quasicrystal Yb$_{15}$Al$_{34}$Au$_{51}$ is shown to be explained from the theory of critical Yb-valence fluctuations starting from the Hamiltonians for describing the low-energy electronic states in both systems.