  The mechanism of the emergence of robust quantum criticality in the
heavy-electron quasicrystal Yb15Al34Au51 is analyzed theoretically. By
constructing a minimal model for the quasicrystal and its crystalline
approximant, which contain concentric shell structures with Yb and Al-Au
clusters, we show that a set of quantum critical points of the first-order
valence transition of Yb appears as spots in the ground-state phase diagram.
Their critical regions overlap each other, giving rise to a wide quantum
critical region. This well explains the robust criticality observed in
Yb15Al34Au51 under pressure, and predicts the emergence of the common
criticality in the crystalline approximant under pressure. The wider critical
region in the quasicrystal than that in the crystalline approximant in the T-P
phase diagram and the field-induced valence-crossover "region" in the T-H phase
diagram are predicted to appear.
