Quantum critical behavior in magnetic quasicrystals and approximant crystals

The electronic states of quasicrystals are believed to be critical, neither extended nor localized. To experimentally establish such the critical state remains a formidable challenge. In the Au-Al-Yb quasicrystal, we observed quantum critical phenomena that are characterized by unconventional critical indices similar to those of Yb-based heavy fermions. In contrast, no divergence was observed in the Au-Al-Yb approximant crystal. These results lead us to suggest that the observed quantum criticality is related to the critical state unique to the quasicrystals. Here we review these results, including the recent observation of the superconductivity in the Tsai-type approximant crystal that is isostructural to the Au-Al-Yb approximant, and argue that the quantum criticality of the quasicrystal results from the combined effect of the quasiperiodicity and the electron correlation.