Berezinskii-Kosterlitz-Thouless Transition and Multifractal Critical Phase in Two-Dimensional Quantum Percolation

Abstract

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-type 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. 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.

Figures (5)

• Figure 1: (a) Average ratio of adjacent energy level spacings, $\langle r \rangle$, as a function of the site occupation probability $p$, for different lattice sizes $L$ and fixed energy interval, $E$. Dashed and dotted horizontal lines respectively correspond to $\langle r\rangle$ for Poisson and Wigner-Dyson distributions. Dashed vertical lines indicate our estimates for the thresholds $p_c$ and $p_q$. (b) Same as (a) but for $d\langle r\rangle/dp$. (c) Finite-size scaling plot of (a). (d) $\chi^2$ minimization to determine the optimal $p_q$ and $\nu$ (red cross) used in (c). When not shown, error bars are smaller than data points.
• Figure 2: Size dependence of the participation entropy ${S}_2$, for several occupation probabilities $p$, ranging across the different regimes appearing in Fig. \ref{['fig:ratio_of_gaps']}; see text. The colorbar encodes the mean adjacent–gap ratio $\langle r\rangle$. Dashed lines are guides to the eye.
• Figure 3: (a) Participation entropy as a function of $p$ for different system sizes $L$. The inset shows the data collapse using a power-law correlation length. (b) Derivative $dS_{2}^{p}/dp$ as a function of $p$. The size-invariant crossing points are marked by the vertical line near $p_c = 0.5927$. (c) Data collapse of $dS_{2}/dp$ near $p_c$, assuming an exponential BKT-like localization length. The inset shows $\chi^{2}$ as a function of $b$ with the remaining fitting parameters, $p_0$ and $p_1$, fixed. (d) Data collapse of $dS_{2}/dp$ near $p_c$, assuming a power-law correlation length. The inset shows $\chi^{2}$ as a function of $\nu$ with fixed $p_0=0.6$ and $\nu=4/3$.
• Figure 4: (a) Color map of $D_q$ versus $(p,q)$. Dashed vertical lines indicate $p_c$ and $p_q$, $p_c<p_q$. (b) Multifractal spectra $f(\alpha)$ for different values of $p$, as given by the color map on the right hand side.
• Figure 5: Phase diagram in the $(p,E)$ plane. The surface shows the fractal dimension $D_2(p, E)$, while the color encodes the mean adjacent–gap ratio $\langle r\rangle$.