The mechanism of the emergence of robust quantum criticality in theheavy-electron quasicrystal Yb15Al34Au51 is analyzed theoretically. Byconstructing a minimal model for the quasicrystal and its crystallineapproximant, which contain concentric shell structures with Yb and Al-Auclusters, we show that a set of quantum critical points of the first-ordervalence transition of Yb appears as spots in the ground-state phase diagram.Their critical regions overlap each other, giving rise to a wide quantumcritical region. This well explains the robust criticality observed inYb15Al34Au51 under pressure, and predicts the emergence of the commoncriticality in the crystalline approximant under pressure. The wider criticalregion in the quasicrystal than that in the crystalline approximant in the T-Pphase diagram and the field-induced valence-crossover "region" in the T-H phasediagram are predicted to appear.
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