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We study the Ising model in a hierarchical small-world network byrenormalization group analysis, and find a phase transition between an orderedphase and a critical phase, which is driven by the coupling strength of theshortcut edges. Unlike ordinary phase transitions, which are related tounstable renormalization fixed points (FPs), the singularity in the orderedphase of the present model is governed by the FP that coincides with the stableFP of the ordered phase. The weak stability of the FP yields peculiarcriticalities including logarithmic behavior. On the other hand, the criticalphase is related to a nontrivial FP, which depends on the coupling strength andis continuously connected to the ordered FP at the transition point. We showthat this continuity indicates the existence of a finitecorrelation-length-like quantity inside the critical phase, which diverges uponapproaching the transition point.

Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network