Quantum percolation on Lieb Lattices

TL;DR

This study addresses localization in quantum percolation with off-diagonal disorder on Lieb lattices in 2D and 3D. It combines level-spacing statistics from Random Matrix Theory with finite-size scaling and cluster selection via the Hoshen-Kopelman algorithm to extract the quantum percolation thresholds $p_q$ and correlation-length exponents $\nu$ for both site and bond percolation. The main findings show that site and bond QP share a universality class, $\nu$ decreases with dimensionality, and in 3D the QP on Lieb lattices falls into the Anderson impurity model (AIM) universality class; thresholds obey a hierarchy consistent with classical percolation and depend on average coordination. These results clarify how lattice geometry and dimensionality shape quantum transport in disordered flat-band systems and have implications for ultracold-atom experiments and correlated-electronic materials where off-diagonal disorder is controllable.

Abstract

We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory, and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with $ν$ decreasing with lattice dimensionality, $d$, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.

Figures (7)

• Figure 1: The two-dimensional Lieb lattice (a), and its three-dimensional extensions: (b) the layered Lieb lattice, LLL, and (c) the perovskite lattice, PL.
• Figure 2: Density of states, $N(E)$, for different site occupation probabilities for (a) square, and (b) simple cubic lattices, with linear sizes $L = 34$ and $L = 15$, respectively; results for bond disorder are similar. The peaks are contributions from small isolated clusters; see text.
• Figure 3: The level spacing distribution for different concentrations of active sites for the square (a) and the simple cubic (b) lattices, respectively with linear sizes $L = 34$ and $L = 15$ . For comparison, we also show analytical expressions for the gaussian orthogonal ensemble distribution (GOE) and the Poisson distribution.
• Figure 4: The function $\gamma (p,L)$, for different system sizes and occupation probabilities, for the square (a) and simple cubic (d) lattices on the quantum site percolation. The chi-squared values of a polynomial fit to the data collapse is shown in (b) for the square lattice and (e) for the simple cubic lattice, for fixed $p_q$. The optimal data collapse is shown in (c) and (f) for the square and simple cubic lattice respectively.
• Figure 5: The function $\gamma (p,L)$ [Eq. \ref{['Eq.gamma_scaling']}] for the two-dimensional Lieb lattice, and for different system sizes: (a) site and (b) bond QP. The insets show the data collapse according to Eq. \ref{['eq. correlationlenght']}