  We study the Ising model in a hierarchical small-world network by
renormalization group analysis, and find a phase transition between an ordered
phase and a critical phase, which is driven by the coupling strength of the
shortcut edges. Unlike ordinary phase transitions, which are related to
unstable renormalization fixed points (FPs), the singularity in the ordered
phase of the present model is governed by the FP that coincides with the stable
FP of the ordered phase. The weak stability of the FP yields peculiar
criticalities including logarithmic behavior. On the other hand, the critical
phase is related to a nontrivial FP, which depends on the coupling strength and
is continuously connected to the ordered FP at the transition point. We show
that this continuity indicates the existence of a finite
correlation-length-like quantity inside the critical phase, which diverges upon
approaching the transition point.
