  To get insight into the mechanism of emergence of unconventional quantum
criticality observed in quasicrystal Yb$_{15}$Al$_{34}$Au$_{51}$, the
approximant crystal Yb$_{14}$Al$_{35}$Au$_{51}$ is analyzed theoretically. By
constructing a minimal model for the approximant crystal, the heavy
quasiparticle band is shown to emerge near the Fermi level because of strong
correlation of 4f electrons at Yb. We find that charge-transfer mode between 4f
electron at Yb on the 3rd shell and 3p electron at Al on the 4th shell in
Tsai-type cluster is considerably enhanced with almost flat momentum
dependence. The mode-coupling theory shows that magnetic as well as valence
susceptibility exhibits $\chi\sim T^{-0.5}$ for zero-field limit and is
expressed as a single scaling function of the ratio of temperature to magnetic
field $T/B$ over four decades even in the approximant crystal when some
condition is satisfied by varying parameters, e.g., by applying pressure. The
key origin is clarified to be due to the strong locality of the critical
Yb-valence fluctuation and small Brillouin zone reflecting the large unit cell,
giving rise to the extremely-small characteristic energy scale. This also gives
a natural explanation for the quantum criticality in the quasicrystal
corresponding to the infinite limit of the unit-cell size.
