We present a numerical study of the two-dimensional quantum percolation model, revealing that a critical region with multifractal eigenstates mediates the transition from localized to delocalized states. By analyzing the mean level ratio and participation entropy, we identify two distinct transitions: a Berezinskii-Kosterlitz-Thouless (BKT) transition at the classical percolation threshold, separating the localized and critical phases, and a power-law-like transition at a larger concentration, marking the onset of full delocalization. The critical phase is characterized by multifractal eigenstates, as evidenced by the generalized fractal dimension and multifractal spectrum. We present a detailed discussion of the results for a spectral window $E/t=0.60\pm 0.05$. The analyses were repeated for energy windows centered at different values of $E/t$, and the results are displayed in the form of a phase diagram as a function of energy; the outcome highlights the robustness of both the multifractal phase and the BTK transition. Altogether, our results establish that in the marginal two-dimensional case, the Anderson impurity model and the quantum percolation model belong to different universality classes.

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• Percolation
• Phase transitions
• Quantum transport
• Spinless fermions
• Disordered systems