Percolation on Lieb lattices

TL;DR

This paper investigates percolation thresholds and critical phenomena on Lieb lattices using Monte Carlo simulations, providing key geometrical parameters relevant for understanding emergent electronic states in these systems.

Contribution

It offers the first detailed estimates of percolation thresholds and correlation length exponents for Lieb lattices, enhancing understanding of their geometrical critical properties.

Findings

Percolation thresholds follow a mean-field trend with coordination number.

Correlation length exponents align with known universality classes.

Threshold estimates aid in studying electronic phenomena in Lieb lattices.

Abstract

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 (LLL) or to a perovskite lattice (PL). 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 Lieb lattices 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.