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arXiv:2409.04610 · Physical review. E

Quantum percolation on Lieb Lattices

W. S. Oliveira, J.P. de Lima, Raimundo R. dos Santos

Published Mar 13, 2025• 5 citations
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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.

DOI
10.1103/physreve.111.034117

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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.
We study site- and bond-percolation on a class of lattices referred to as Lieb lattices. In two dimensions the Lieb lattice (LL) is also known as the decorated square lattice, or as the CuO_{2} lattice; in three dimensions it can be generalized to a layered Lieb lattice or to a perovskite lattice. Emergent electronic phenomena, such as topological states and ferrimagnetism, have been predicted to occur in these systems, which may be realized in optical lattices as well as in solid state. Since the study of the interplay between quantum fluctuations and disorder in these systems requires the availability of accurate estimates of geometrical critical parameters, such as percolation thresholds and correlation length exponents, here we use Monte Carlo simulations to obtain these data for LLs when a site (or bond) is present with probability p. We have found that the thresholds satisfy a mean-field (Bethe lattice) trend, namely that the critical concentration, p_{c}, increases as the average coordination number decreases; our estimates for the correlation length exponent are in line with the expectation that there is no change in the universality class.
In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold pq{p}_{q}pq​. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of pq{p}_{q}pq​ and the lowest value ${p}_{q}\ensuremath{\simeq}0.665$ are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents $\ensuremath{\nu}$ not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold pq<1{p}_{q}<1pq​<1 in the two-dimensional quantum percolation problem.
The quantum site and bond percolation problem, which is defined by a disordered tight-binding Hamiltonian with a binary probability distribution, is studied using finite-size-scaling methods. For the simple square lattice, we find that all states are exponentially localized for any amount of disorder, in agreement with the scalng theory of localization and in disagreement with recent claims of a localization transition in two dimensions. The localization length \ensuremath{\lambda} is given by exp{B[p/(1-p)]y{]}^{\mathit{y}}]y} with y very close to 0.5 and p the probability that a site or a bond is present.
We investigate quantum percolation in a honeycomb lattice with site dilution and random spin-orbit coupling. Using exact diagonalization combined with finite-size scaling analysis, we study the metal-insulator transition, extracting the quantum percolation threshold pqp_qpq​, and the correlation-length exponent, ννν. In the absence of spin-orbit coupling, we find that pqp_qpq​ remains finite and demonstrate that the quantum threshold is significantly higher than the classical site-percolation threshold pcp_cpc​ of the honeycomb lattice. When spin-orbit coupling is present, the spectral statistics exhibit a crossover from the Gaussian orthogonal ensemble to the Gaussian symplectic ensemble, reflecting the change in symmetry class. Simultaneously, the quantum percolation threshold shifts systematically to lower occupation probabilities, indicating that the spin-orbit coupling favors delocalization. For sufficiently strong spin-orbit coupling, pqp_qpq​ tends to saturate, while the critical exponent approaches the expected one of the two-dimensional symplectic universality class.
We investigate quantum percolation in a honeycomb lattice with site dilution and random spin-orbit coupling. Using exact diagonalization combined with finite-size scaling analysis, we study the metal-insulator transition, extracting the quantum percolation threshold p_{q}, and the correlation-length exponent ν. In the absence of spin-orbit coupling, we find that p_{q} remains finite and demonstrate that the quantum threshold is significantly higher than the classical site-percolation threshold p_{c} of the honeycomb lattice. When spin-orbit coupling is present, the spectral statistics exhibit a crossover from the Gaussian orthogonal ensemble to the Gaussian symplectic ensemble, reflecting the change in symmetry class. Simultaneously, the quantum percolation threshold shifts systematically to lower occupation probabilities, indicating that the spin-orbit coupling favors delocalization. For sufficiently strong spin-orbit coupling, p_{q} tends to saturate, while the critical exponent approaches the expected one of the two-dimensional symplectic universality class.
We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.