Berezinskii-Kosterlitz-Thouless-type Transition in Site Percolation on the Diamond Hierarchical Lattice

We study site percolation on the diamond hierarchical lattice, a finite-dimensional fractal network, using an exact generating-function analysis. In contrast to bond percolation, site percolation on this lattice does not undergo a transition from a nonpercolating phase to a percolating phase. Instead, the system exhibits a nonpercolating phase for $p<p_{\rm c}$ and a critical phase for $p>p_{\rm c}$. In the critical phase, the size of the largest cluster remains subextensive, scaling as $N^{\psi(p)}$, where the fractal exponent $\psi(p)$ varies continuously with $p$. By analyzing the renormalization-group recursion relation in the vicinity of $p_{\rm c}$, we show that the correlation length exhibits a Berezinskii-Kosterlitz-Thouless-type essential singularity, $\xi(p)\sim \exp \left({\rm const}/\sqrt{p_{\rm c}-p}\right)$ for $p \to p_{\rm c}^-$, which is further confirmed by finite-size scaling analyses showing excellent data collapse. These results demonstrate that critical phases in percolation can emerge even on finite-dimensional networks and that exponential volume growth is not necessary for such phases to appear. We argue that the critical phase on the diamond hierarchical lattice stems from site dilution remaining relevant under renormalization.