Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network

We study bond percolation on a one-parameter family of hierarchical small-world network, and find a meta-transition between the inverted BKT transition and the abrupt transition driven by changing the network topology. It is found that the order parameter is continuous and fractal exponent is discontinuous in the inverted BKT transition, and oppositely, the former is discontinuous and the latter is continuous in the abrupt transition. The gaps of the order parameter and fractal exponent in each transition go to vanish as approaching the meta-transition point. This point corresponds to a marginal power-law transition. In the renormalization group formalism, this meta-transition corresponds to the transition between transcritical and saddle-node bifurcations of the fixed point via a pitchfork bifurcation.